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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, the `!modular lambda`! function λ(τ)`:cite-ref-1[`F5bf`_`[note 1`#cite-note-1]`_`f] is a highly symmetric `F33f`_`[holomorphic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holomorphic_function]`_`f on the complex `F33f`_`[upper half-plane`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Upper_half-plane]`_`f. It is invariant under the fractional linear action of the `F33f`_`[congruence group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Congruence_subgroup]`_`f Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the `F33f`_`[modular curve`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modular_curve]`_`f `*X`*(2). Over any point τ, its value can be described as a `F33f`_`[cross ratio`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cross_ratio]`_`f of the branch points of a ramified double cover of the projective line by the `F33f`_`[elliptic curve`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_curve]`_`f C / ⟨ ⟨ 1 , τ τ ⟩ ⟩ {\\displaystyle \\mathbb {C} /\\langle 1,\\tau \\rangle } , where the map is defined as the quotient by the [−1] involution.

The q-expansion, where q = e π π i τ τ {\\displaystyle q=e^{\\pi i\\tau }} is the `F33f`_`[nome`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nome_(mathematics)]`_`f, is given by:

λ λ ( τ τ ) = 16 q − − 128 q 2 + 704 q 3 − − 3072 q 4 + 11488 q 5 − − 38400 q 6 + … … {\\displaystyle \\lambda (\\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\\dots } . `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A115977

By symmetrizing the lambda function under the canonical action of the symmetric group `*S`*3 on `*X`*(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group SL 2 ⁡ ⁡ ( Z ) {\\displaystyle \\operatorname {SL} _{2}(\\mathbb {Z} )} , and it is in fact Klein's modular `F33f`_`[j-invariant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J-invariant]`_`f.

>>Contents

• `F0af`_`[Modular properties`#modular-properties]`_`f
• `F0af`_`[Relations to other functions`#relations-to-other-functions]`_`f
• `F0af`_`[Modular equations`#modular-equations]`_`f
• `F0af`_`[Lambda-star`#lambda-star]`_`f
• `F0af`_`[Definition and computation of lambda-star`#definition-and-computation-of-lambda-star]`_`f
• `F0af`_`[Properties of lambda-star`#properties-of-lambda-star]`_`f
• `F0af`_`[Ramanujan's class invariants`#ramanujan-s-class-invariants]`_`f
• `F0af`_`[Other appearances`#other-appearances]`_`f
• `F0af`_`[Little Picard theorem`#little-picard-theorem]`_`f
• `F0af`_`[Moonshine`#moonshine]`_`f
• `F0af`_`[Footnotes`#footnotes]`_`f
• `F0af`_`[References`#references]`_`f
• `F0af`_`[Notes`#notes]`_`f
• `F0af`_`[Other`#other]`_`f
• `F0af`_`[External links`#external-links]`_`f

-─

>>Modular properties

The function λ λ ( τ τ ) {\\displaystyle \\lambda (\\tau )} is invariant under the group generated by`:cite-ref-c115-2-0[`F5bf`_`[1`#cite-note-c115-2]`_`f]

τ τ ↦ ↦ τ τ + 2 ; τ τ ↦ ↦ τ τ 1 − − 2 τ τ . {\\displaystyle \\tau \\mapsto \\tau +2\\ ;\\ \\tau \\mapsto {\\frac {\\tau }{1-2\\tau }}\\ .}

The generators of the modular group act by`:cite-ref-c109-3-0[`F5bf`_`[2`#cite-note-c109-3]`_`f]

τ τ ↦ ↦ τ τ + 1 : λ λ ↦ ↦ λ λ λ λ − − 1 ; {\\displaystyle \\tau \\mapsto \\tau +1\\ :\\ \\lambda \\mapsto {\\frac {\\lambda }{\\lambda -1}}\\,;}
τ τ ↦ ↦ − − 1 τ τ : λ λ ↦ ↦ 1 − − λ λ . {\\displaystyle \\tau \\mapsto -{\\frac {1}{\\tau }}\\ :\\ \\lambda \\mapsto 1-\\lambda \\ .}

Consequently, the action of the modular group on λ λ ( τ τ ) {\\displaystyle \\lambda (\\tau )} is that of the `F33f`_`[anharmonic group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Anharmonic_group]`_`f, giving the six values of the `F33f`_`[cross-ratio`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cross-ratio]`_`f:`:cite-ref-c110-4-0[`F5bf`_`[3`#cite-note-c110-4]`_`f]

{ λ λ , 1 1 − − λ λ , λ λ − − 1 λ λ , 1 λ λ , λ λ λ λ − − 1 , 1 − − λ λ } . {\\displaystyle \\left\\lbrace {\\lambda ,{\\frac {1}{1-\\lambda }},{\\frac {\\lambda -1}{\\lambda }},{\\frac {1}{\\lambda }},{\\frac {\\lambda }{\\lambda -1}},1-\\lambda }\\right\\rbrace \\ .}

>>Relations to other functions

It is the `F33f`_`[square`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square_(algebra)]`_`f of the elliptic modulus,`:cite-ref-c108-5-0[`F5bf`_`[4`#cite-note-c108-5]`_`f] that is, λ λ ( τ τ ) = k 2 ( τ τ ) {\\displaystyle \\lambda (\\tau )=k^{2}(\\tau )} . In terms of the `F33f`_`[Dedekind eta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dedekind_eta_function]`_`f η η ( τ τ ) {\\displaystyle \\eta (\\tau )} and `F33f`_`[theta functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theta_function]`_`f,`:cite-ref-c108-5-1[`F5bf`_`[4`#cite-note-c108-5]`_`f]

λ λ ( τ τ ) = ( 2 η η ( τ τ 2 ) η η 2 ( 2 τ τ ) η η 3 ( τ τ ) ) 8 = 16 ( η η ( τ τ / 2 ) η η ( 2 τ τ ) ) 8 + 16 = θ θ 2 4 ( τ τ ) θ θ 3 4 ( τ τ ) {\\displaystyle \\lambda (\\tau )={\\Bigg (}{\\frac {{\\sqrt {2}}\\,\\eta ({\\tfrac {\\tau }{2}})\\eta ^{2}(2\\tau )}{\\eta ^{3}(\\tau )}}{\\Bigg )}^{8}={\\frac {16}{\\left({\\frac {\\eta (\\tau /2)}{\\eta (2\\tau )}}\\right)^{8}+16}}={\\frac {\\theta _{2}^{4}(\\tau )}{\\theta _{3}^{4}(\\tau )}}}

and,

1 ( λ λ ( τ τ ) ) 1 / 4 − − ( λ λ ( τ τ ) ) 1 / 4 = 1 2 ( η η ( τ τ 4 ) η η ( τ τ ) ) 4 = 2 θ θ 4 2 ( τ τ 2 ) θ θ 2 2 ( τ τ 2 ) {\\displaystyle {\\frac {1}{{\\big (}\\lambda (\\tau ){\\big )}^{1/4}}}-{\\big (}\\lambda (\\tau ){\\big )}^{1/4}={\\frac {1}{2}}\\left({\\frac {\\eta ({\\tfrac {\\tau }{4}})}{\\eta (\\tau )}}\\right)^{4}=2\\,{\\frac {\\theta _{4}^{2}({\\tfrac {\\tau }{2}})}{\\theta _{2}^{2}({\\tfrac {\\tau }{2}})}}}

where`:cite-ref-c63-6-0[`F5bf`_`[5`#cite-note-c63-6]`_`f]

θ θ 2 ( τ τ ) = ∑ ∑ n = − − ∞ ∞ ∞ ∞ e π π i τ τ ( n + 1 / 2 ) 2 {\\displaystyle \\theta _{2}(\\tau )=\\sum _{n=-\\infty }^{\\infty }e^{\\pi i\\tau (n+1/2)^{2}}}

θ θ 3 ( τ τ ) = ∑ ∑ n = − − ∞ ∞ ∞ ∞ e π π i τ τ n 2 {\\displaystyle \\theta _{3}(\\tau )=\\sum _{n=-\\infty }^{\\infty }e^{\\pi i\\tau n^{2}}}

θ θ 4 ( τ τ ) = ∑ ∑ n = − − ∞ ∞ ∞ ∞ ( − − 1 ) n e π π i τ τ n 2 {\\displaystyle \\theta _{4}(\\tau )=\\sum _{n=-\\infty }^{\\infty }(-1)^{n}e^{\\pi i\\tau n^{2}}}

In terms of the half-periods of `F33f`_`[Weierstrass's elliptic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weierstrass's_elliptic_functions]`_`f, let [ ω ω 1 , ω ω 2 ] {\\displaystyle [\\omega _{1},\\omega _{2}]} be a `F33f`_`[fundamental pair of periods`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fundamental_pair_of_periods]`_`f with τ τ = ω ω 2 ω ω 1 {\\displaystyle \\tau ={\\frac {\\omega _{2}}{\\omega _{1}}}} .

e 1 = ℘ ℘ ( ω ω 1 2 ) , e 2 = ℘ ℘ ( ω ω 2 2 ) , e 3 = ℘ ℘ ( ω ω 1 + ω ω 2 2 ) {\\displaystyle e_{1}=\\wp \\left({\\frac {\\omega _{1}}{2}}\\right),\\quad e_{2}=\\wp \\left({\\frac {\\omega _{2}}{2}}\\right),\\quad e_{3}=\\wp \\left({\\frac {\\omega _{1}+\\omega _{2}}{2}}\\right)}

we have`:cite-ref-c108-5-2[`F5bf`_`[4`#cite-note-c108-5]`_`f]

λ λ = e 3 − − e 2 e 1 − − e 2 . {\\displaystyle \\lambda ={\\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\\,.}

Since the three half-period values are distinct, this shows that λ λ {\\displaystyle \\lambda } does not take the value 0 or 1.`:cite-ref-c108-5-3[`F5bf`_`[4`#cite-note-c108-5]`_`f]

The relation to the `F33f`_`[j-invariant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J-invariant]`_`f is`:cite-ref-c117-7-0[`F5bf`_`[6`#cite-note-c117-7]`_`f]`:cite-ref-8[`F5bf`_`[7`#cite-note-8]`_`f]

j ( τ τ ) = 256 ( 1 − − λ λ ( 1 − − λ λ ) ) 3 ( λ λ ( 1 − − λ λ ) ) 2 = 256 ( 1 − − λ λ + λ λ 2 ) 3 λ λ 2 ( 1 − − λ λ ) 2 . {\\displaystyle j(\\tau )={\\frac {256(1-\\lambda (1-\\lambda ))^{3}}{(\\lambda (1-\\lambda ))^{2}}}={\\frac {256(1-\\lambda +\\lambda ^{2})^{3}}{\\lambda ^{2}(1-\\lambda )^{2}}}\\ .}

which is the `*j`*-invariant of the elliptic curve of `F33f`_`[Legendre form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Legendre_form]`_`f y 2 = x ( x − − 1 ) ( x − − λ λ ) {\\displaystyle y^{2}=x(x-1)(x-\\lambda )}

Given m ∈ ∈ C ∖ ∖ { 0 , 1 } {\\displaystyle m\\in \\mathbb {C} \\setminus \\{0,1\\}} , let

τ τ = i K { 1 − − m } K { m } {\\displaystyle \\tau =i{\\frac {K\\{1-m\\}}{K\\{m\\}}}}

where K {\\displaystyle K} is the `F33f`_`[complete elliptic integral of the first kind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_integral]`_`f with parameter m = k 2 {\\displaystyle m=k^{2}} . Then

λ λ ( τ τ ) = m . {\\displaystyle \\lambda (\\tau )=m.}

>>Modular equations

The `*modular equation of degree`* p {\\displaystyle p} (where p {\\displaystyle p} is a prime number) is an algebraic equation in λ λ ( p τ τ ) {\\displaystyle \\lambda (p\\tau )} and λ λ ( τ τ ) {\\displaystyle \\lambda (\\tau )} . If λ λ ( p τ τ ) = u 8 {\\displaystyle \\lambda (p\\tau )=u^{8}} and λ λ ( τ τ ) = v 8 {\\displaystyle \\lambda (\\tau )=v^{8}} , the modular equations of degrees p = 2 , 3 , 5 , 7 {\\displaystyle p=2,3,5,7} are, respectively,`:cite-ref-9[`F5bf`_`[8`#cite-note-9]`_`f]

( 1 + u 4 ) 2 v 8 − − 4 u 4 = 0 , {\\displaystyle (1+u^{4})^{2}v^{8}-4u^{4}=0,}
u 4 − − v 4 + 2 u v ( 1 − − u 2 v 2 ) = 0 , {\\displaystyle u^{4}-v^{4}+2uv(1-u^{2}v^{2})=0,}
u 6 − − v 6 + 5 u 2 v 2 ( u 2 − − v 2 ) + 4 u v ( 1 − − u 4 v 4 ) = 0 , {\\displaystyle u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=0,}
( 1 − − u 8 ) ( 1 − − v 8 ) − − ( 1 − − u v ) 8 = 0. {\\displaystyle (1-u^{8})(1-v^{8})-(1-uv)^{8}=0.}

The quantity v {\\displaystyle v} (and hence u {\\displaystyle u} ) can be thought of as a `F33f`_`[holomorphic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holomorphic_function]`_`f on the upper half-plane Im ⁡ ⁡ τ τ > 0 {\\displaystyle \\operatorname {Im} \\tau >0} :

v = ∏ ∏ k = 1 ∞ ∞ tanh ⁡ ⁡ ( k − − 1 / 2 ) π π i τ τ = 2 e π π i τ τ / 8 ∑ ∑ k ∈ ∈ Z e ( 2 k 2 + k ) π π i τ τ ∑ ∑ k ∈ ∈ Z e k 2 π π i τ τ = 2 e π π i τ τ / 8 1 + e π π i τ τ 1 + e π π i τ τ + e 2 π π i τ τ 1 + e 2 π π i τ τ + e 3 π π i τ τ 1 + e 3 π π i τ τ + ⋱ ⋱ {\\displaystyle {\\begin{aligned}v&=\\prod _{k=1}^{\\infty }\\tanh {\\frac {(k-1/2)\\pi i}{\\tau }}={\\sqrt {2}}e^{\\pi i\\tau /8}{\\frac {\\sum _{k\\in \\mathbb {Z} }e^{(2k^{2}+k)\\pi i\\tau }}{\\sum _{k\\in \\mathbb {Z} }e^{k^{2}\\pi i\\tau }}}\\\\&={\\cfrac {{\\sqrt {2}}e^{\\pi i\\tau /8}}{1+{\\cfrac {e^{\\pi i\\tau }}{1+e^{\\pi i\\tau }+{\\cfrac {e^{2\\pi i\\tau }}{1+e^{2\\pi i\\tau }+{\\cfrac {e^{3\\pi i\\tau }}{1+e^{3\\pi i\\tau }+\\ddots }}}}}}}}\\end{aligned}}}

Since λ λ ( i ) = 1 / 2 {\\displaystyle \\lambda (i)=1/2} , the modular equations can be used to give `F33f`_`[algebraic values`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_number]`_`f of λ λ ( p i ) {\\displaystyle \\lambda (pi)} for any prime p {\\displaystyle p} .`:cite-ref-10[`F5bf`_`[note 2`#cite-note-10]`_`f] The algebraic values of λ λ ( n i ) {\\displaystyle \\lambda (ni)} are also given by`:cite-ref-jacobi-11-0[`F5bf`_`[9`#cite-note-jacobi-11]`_`f]`:cite-ref-12[`F5bf`_`[note 3`#cite-note-12]`_`f]

λ λ ( n i ) = ∏ ∏ k = 1 n / 2 sl 8 ⁡ ⁡ ( 2 k − − 1 ) ϖ ϖ 2 n ( n even ) {\\displaystyle \\lambda (ni)=\\prod _{k=1}^{n/2}\\operatorname {sl} ^{8}{\\frac {(2k-1)\\varpi }{2n}}\\quad (n\\,{\\text{even}})}
λ λ ( n i ) = 1 2 n ∏ ∏ k = 1 n − − 1 ( 1 − − sl 2 ⁡ ⁡ k ϖ ϖ n ) 2 ( n odd ) {\\displaystyle \\lambda (ni)={\\frac {1}{2^{n}}}\\prod _{k=1}^{n-1}\\left(1-\\operatorname {sl} ^{2}{\\frac {k\\varpi }{n}}\\right)^{2}\\quad (n\\,{\\text{odd}})}

where sl {\\displaystyle \\operatorname {sl} } is the `F33f`_`[lemniscate sine`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lemniscate_elliptic_functions]`_`f and ϖ ϖ {\\displaystyle \\varpi } is the `F33f`_`[lemniscate constant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lemniscate_constant]`_`f.

>>Lambda-star

>>>Definition and computation of lambda-star

The function λ λ ∗ ∗ ( x ) {\\displaystyle \\lambda ^{*}(x)} `:cite-ref-13[`F5bf`_`[10`#cite-note-13]`_`f] (where x ∈ ∈ R + {\\displaystyle x\\in \\mathbb {R} ^{+}} ) gives the value of the elliptic modulus k {\\displaystyle k} , for which the `F33f`_`[complete elliptic integral of the first kind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_integral]`_`f K ( k ) {\\displaystyle K(k)} and its complementary counterpart K ( 1 − − k 2 ) {\\displaystyle K({\\sqrt {1-k^{2}}})} are related by following expression:

K [ 1 − − λ λ ∗ ∗ ( x ) 2 ] K [ λ λ ∗ ∗ ( x ) ] = x {\\displaystyle {\\frac {K\\left[{\\sqrt {1-\\lambda ^{*}(x)^{2}}}\\right]}{K[\\lambda ^{*}(x)]}}={\\sqrt {x}}}

The values of λ λ ∗ ∗ ( x ) {\\displaystyle \\lambda ^{*}(x)} can be computed as follows:

λ λ ∗ ∗ ( x ) = θ θ 2 2 ( i x ) θ θ 3 2 ( i x ) {\\displaystyle \\lambda ^{*}(x)={\\frac {\\theta _{2}^{2}(i{\\sqrt {x}})}{\\theta _{3}^{2}(i{\\sqrt {x}})}}}

λ λ ∗ ∗ ( x ) = [ ∑ ∑ a = − − ∞ ∞ ∞ ∞ exp ⁡ ⁡ [ − − ( a + 1 / 2 ) 2 π π x ] ] 2 [ ∑ ∑ a = − − ∞ ∞ ∞ ∞ exp ⁡ ⁡ ( − − a 2 π π x ) ] − − 2 {\\displaystyle \\lambda ^{*}(x)=\\left[\\sum _{a=-\\infty }^{\\infty }\\exp[-(a+1/2)^{2}\\pi {\\sqrt {x}}]\\right]^{2}\\left[\\sum _{a=-\\infty }^{\\infty }\\exp(-a^{2}\\pi {\\sqrt {x}})\\right]^{-2}}

λ λ ∗ ∗ ( x ) = [ ∑ ∑ a = − − ∞ ∞ ∞ ∞ sech ⁡ ⁡ [ ( a + 1 / 2 ) π π x ] ] [ ∑ ∑ a = − − ∞ ∞ ∞ ∞ sech ⁡ ⁡ ( a π π x ) ] − − 1 {\\displaystyle \\lambda ^{*}(x)=\\left[\\sum _{a=-\\infty }^{\\infty }\\operatorname {sech} [(a+1/2)\\pi {\\sqrt {x}}]\\right]\\left[\\sum _{a=-\\infty }^{\\infty }\\operatorname {sech} (a\\pi {\\sqrt {x}})\\right]^{-1}}

The functions λ λ ∗ ∗ {\\displaystyle \\lambda ^{*}} and λ λ {\\displaystyle \\lambda } are related to each other in this way:

λ λ ∗ ∗ ( x ) = λ λ ( i x ) {\\displaystyle \\lambda ^{*}(x)={\\sqrt {\\lambda (i{\\sqrt {x}})}}}

>>>Properties of lambda-star

Every λ λ ∗ ∗ {\\displaystyle \\lambda ^{*}} value of a positive `F33f`_`[rational number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_number]`_`f is a positive `F33f`_`[algebraic number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_number]`_`f:

λ λ ∗ ∗ ( x ∈ ∈ Q + ) ∈ ∈ A + . {\\displaystyle \\lambda ^{*}(x\\in \\mathbb {Q} ^{+})\\in \\mathbb {A} ^{+}.}

K ( λ λ ∗ ∗ ( x ) ) {\\displaystyle K(\\lambda ^{*}(x))} and E ( λ λ ∗ ∗ ( x ) ) {\\displaystyle E(\\lambda ^{*}(x))} (the `F33f`_`[complete elliptic integral of the second kind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_integral]`_`f) can be expressed in closed form in terms of the `F33f`_`[gamma function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gamma_function]`_`f for any x ∈ ∈ Q + {\\displaystyle x\\in \\mathbb {Q} ^{+}} , as Selberg and Chowla proved in 1949.`:cite-ref-14[`F5bf`_`[11`#cite-note-14]`_`f]`:cite-ref-15[`F5bf`_`[12`#cite-note-15]`_`f]

The following expression is valid for all n ∈ ∈ N {\\displaystyle n\\in \\mathbb {N} } :

n = ∑ ∑ a = 1 n dn ⁡ ⁡ [ 2 a n K [ λ λ ∗ ∗ ( 1 n ) ] ; λ λ ∗ ∗ ( 1 n ) ] {\\displaystyle {\\sqrt {n}}=\\sum _{a=1}^{n}\\operatorname {dn} \\left[{\\frac {2a}{n}}K\\left[\\lambda ^{*}\\left({\\frac {1}{n}}\\right)\\right];\\lambda ^{*}\\left({\\frac {1}{n}}\\right)\\right]}

where dn {\\displaystyle \\operatorname {dn} } is the `F33f`_`[Jacobi elliptic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jacobi_elliptic_function]`_`f delta amplitudinis with modulus k {\\displaystyle k} .

By knowing one λ λ ∗ ∗ {\\displaystyle \\lambda ^{*}} value, this formula can be used to compute related λ λ ∗ ∗ {\\displaystyle \\lambda ^{*}} values:`:cite-ref-jacobi-11-1[`F5bf`_`[9`#cite-note-jacobi-11]`_`f]

λ λ ∗ ∗ ( n 2 x ) = λ λ ∗ ∗ ( x ) n ∏ ∏ a = 1 n sn ⁡ ⁡ { 2 a − − 1 n K [ λ λ ∗ ∗ ( x ) ] ; λ λ ∗ ∗ ( x ) } 2 {\\displaystyle \\lambda ^{*}(n^{2}x)=\\lambda ^{*}(x)^{n}\\prod _{a=1}^{n}\\operatorname {sn} \\left\\{{\\frac {2a-1}{n}}K[\\lambda ^{*}(x)];\\lambda ^{*}(x)\\right\\}^{2}}

where n ∈ ∈ N {\\displaystyle n\\in \\mathbb {N} } and sn {\\displaystyle \\operatorname {sn} } is the Jacobi elliptic function sinus amplitudinis with modulus k {\\displaystyle k} .

Further relations:

λ λ ∗ ∗ ( x ) 2 + λ λ ∗ ∗ ( 1 / x ) 2 = 1 {\\displaystyle \\lambda ^{*}(x)^{2}+\\lambda ^{*}(1/x)^{2}=1}

[ λ λ ∗ ∗ ( x ) + 1 ] [ λ λ ∗ ∗ ( 4 / x ) + 1 ] = 2 {\\displaystyle [\\lambda ^{*}(x)+1][\\lambda ^{*}(4/x)+1]=2}

λ λ ∗ ∗ ( 4 x ) = 1 − − 1 − − λ λ ∗ ∗ ( x ) 2 1 + 1 − − λ λ ∗ ∗ ( x ) 2 = tan ⁡ ⁡ { 1 2 arcsin ⁡ ⁡ [ λ λ ∗ ∗ ( x ) ] } 2 {\\displaystyle \\lambda ^{*}(4x)={\\frac {1-{\\sqrt {1-\\lambda ^{*}(x)^{2}}}}{1+{\\sqrt {1-\\lambda ^{*}(x)^{2}}}}}=\\tan \\left\\{{\\frac {1}{2}}\\arcsin[\\lambda ^{*}(x)]\\right\\}^{2}}

λ λ ∗ ∗ ( x ) − − λ λ ∗ ∗ ( 9 x ) = 2 [ λ λ ∗ ∗ ( x ) λ λ ∗ ∗ ( 9 x ) ] 1 / 4 − − 2 [ λ λ ∗ ∗ ( x ) λ λ ∗ ∗ ( 9 x ) ] 3 / 4 {\\displaystyle \\lambda ^{*}(x)-\\lambda ^{*}(9x)=2[\\lambda ^{*}(x)\\lambda ^{*}(9x)]^{1/4}-2[\\lambda ^{*}(x)\\lambda ^{*}(9x)]^{3/4}}

a 6 − − f 6 = 2 a f + 2 a 5 f 5 ( a = [ 2 λ λ ∗ ∗ ( x ) 1 − − λ λ ∗ ∗ ( x ) 2 ] 1 / 12 ) ( f = [ 2 λ λ ∗ ∗ ( 25 x ) 1 − − λ λ ∗ ∗ ( 25 x ) 2 ] 1 / 12 ) a 8 + b 8 − − 7 a 4 b 4 = 2 2 a b + 2 2 a 7 b 7 ( a = [ 2 λ λ ∗ ∗ ( x ) 1 − − λ λ ∗ ∗ ( x ) 2 ] 1 / 12 ) ( b = [ 2 λ λ ∗ ∗ ( 49 x ) 1 − − λ λ ∗ ∗ ( 49 x ) 2 ] 1 / 12 ) a 12 − − c 12 = 2 2 ( a c + a 3 c 3 ) ( 1 + 3 a 2 c 2 + a 4 c 4 ) ( 2 + 3 a 2 c 2 + 2 a 4 c 4 ) ( a = [ 2 λ λ ∗ ∗ ( x ) 1 − − λ λ ∗ ∗ ( x ) 2 ] 1 / 12 ) ( c = [ 2 λ λ ∗ ∗ ( 121 x ) 1 − − λ λ ∗ ∗ ( 121 x ) 2 ] 1 / 12 ) ( a 2 − − d 2 ) ( a 4 + d 4 − − 7 a 2 d 2 ) [ ( a 2 − − d 2 ) 4 − − a 2 d 2 ( a 2 + d 2 ) 2 ] = 8 a d + 8 a 13 d 13 ( a = [ 2 λ λ ∗ ∗ ( x ) 1 − − λ λ ∗ ∗ ( x ) 2 ] 1 / 12 ) ( d = [ 2 λ λ ∗ ∗ ( 169 x ) 1 − − λ λ ∗ ∗ ( 169 x ) 2 ] 1 / 12 ) {\\displaystyle {\\begin{aligned}&a^{6}-f^{6}=2af+2a^{5}f^{5}\\,&\\left(a=\\left[{\\frac {2\\lambda ^{*}(x)}{1-\\lambda ^{*}(x)^{2}}}\\right]^{1/12}\\right)&\\left(f=\\left[{\\frac {2\\lambda ^{*}(25x)}{1-\\lambda ^{*}(25x)^{2}}}\\right]^{1/12}\\right)\\\\&a^{8}+b^{8}-7a^{4}b^{4}=2{\\sqrt {2}}ab+2{\\sqrt {2}}a^{7}b^{7}\\,&\\left(a=\\left[{\\frac {2\\lambda ^{*}(x)}{1-\\lambda ^{*}(x)^{2}}}\\right]^{1/12}\\right)&\\left(b=\\left[{\\frac {2\\lambda ^{*}(49x)}{1-\\lambda ^{*}(49x)^{2}}}\\right]^{1/12}\\right)\\\\&a^{12}-c^{12}=2{\\sqrt {2}}(ac+a^{3}c^{3})(1+3a^{2}c^{2}+a^{4}c^{4})(2+3a^{2}c^{2}+2a^{4}c^{4})\\,&\\left(a=\\left[{\\frac {2\\lambda ^{*}(x)}{1-\\lambda ^{*}(x)^{2}}}\\right]^{1/12}\\right)&\\left(c=\\left[{\\frac {2\\lambda ^{*}(121x)}{1-\\lambda ^{*}(121x)^{2}}}\\right]^{1/12}\\right)\\\\&(a^{2}-d^{2})(a^{4}+d^{4}-7a^{2}d^{2})[(a^{2}-d^{2})^{4}-a^{2}d^{2}(a^{2}+d^{2})^{2}]=8ad+8a^{13}d^{13}\\,&\\left(a=\\left[{\\frac {2\\lambda ^{*}(x)}{1-\\lambda ^{*}(x)^{2}}}\\right]^{1/12}\\right)&\\left(d=\\left[{\\frac {2\\lambda ^{*}(169x)}{1-\\lambda ^{*}(169x)^{2}}}\\right]^{1/12}\\right)\\end{aligned}}}

>>>Ramanujan's class invariants

`F33f`_`[Ramanujan's`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Srinivasa_Ramanujan]`_`f class invariants G n {\\displaystyle G_{n}} and g n {\\displaystyle g_{n}} are defined as`:cite-ref-16[`F5bf`_`[13`#cite-note-16]`_`f]

G n = 2 − − 1 / 4 e π π n / 24 ∏ ∏ k = 0 ∞ ∞ ( 1 + e − − ( 2 k + 1 ) π π n ) , {\\displaystyle G_{n}=2^{-1/4}e^{\\pi {\\sqrt {n}}/24}\\prod _{k=0}^{\\infty }\\left(1+e^{-(2k+1)\\pi {\\sqrt {n}}}\\right),}
g n = 2 − − 1 / 4 e π π n / 24 ∏ ∏ k = 0 ∞ ∞ ( 1 − − e − − ( 2 k + 1 ) π π n ) , {\\displaystyle g_{n}=2^{-1/4}e^{\\pi {\\sqrt {n}}/24}\\prod _{k=0}^{\\infty }\\left(1-e^{-(2k+1)\\pi {\\sqrt {n}}}\\right),}

where n ∈ ∈ Q + {\\displaystyle n\\in \\mathbb {Q} ^{+}} . For such n {\\displaystyle n} , the class invariants are algebraic numbers. For example

g 58 = 5 + 29 2 , g 190 = ( 5 + 2 ) ( 10 + 3 ) . {\\displaystyle g_{58}={\\sqrt {\\frac {5+{\\sqrt {29}}}{2}}},\\quad g_{190}={\\sqrt {({\\sqrt {5}}+2)({\\sqrt {10}}+3)}}.}

Identities with the class invariants include`:cite-ref-17[`F5bf`_`[14`#cite-note-17]`_`f]

G n = G 1 / n , g n = 1 g 4 / n , g 4 n = 2 1 / 4 g n G n . {\\displaystyle G_{n}=G_{1/n},\\quad g_{n}={\\frac {1}{g_{4/n}}},\\quad g_{4n}=2^{1/4}g_{n}G_{n}.}

The class invariants are very closely related to the `F33f`_`[Weber modular functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weber_modular_function]`_`f f {\\displaystyle {\\mathfrak {f}}} and f 1 {\\displaystyle {\\mathfrak {f}}_{1}} . These are the relations between lambda-star and the class invariants:

G n = sin ⁡ ⁡ { 2 arcsin ⁡ ⁡ [ λ λ ∗ ∗ ( n ) ] } − − 1 / 12 = 1 / [ 2 λ λ ∗ ∗ ( n ) 12 1 − − λ λ ∗ ∗ ( n ) 2 24 ] {\\displaystyle G_{n}=\\sin\\{2\\arcsin[\\lambda ^{*}(n)]\\}^{-1/12}=1{\\Big /}\\left[{\\sqrt[{12}]{2\\lambda ^{*}(n)}}{\\sqrt[{24}]{1-\\lambda ^{*}(n)^{2}}}\\right]}

g n = tan ⁡ ⁡ { 2 arctan ⁡ ⁡ [ λ λ ∗ ∗ ( n ) ] } − − 1 / 12 = [ 1 − − λ λ ∗ ∗ ( n ) 2 ] / [ 2 λ λ ∗ ∗ ( n ) ] 12 {\\displaystyle g_{n}=\\tan\\{2\\arctan[\\lambda ^{*}(n)]\\}^{-1/12}={\\sqrt[{12}]{[1-\\lambda ^{*}(n)^{2}]/[2\\lambda ^{*}(n)]}}}

λ λ ∗ ∗ ( n ) = tan ⁡ ⁡ { 1 2 arctan ⁡ ⁡ [ g n − − 12 ] } = g n 24 + 1 − − g n 12 {\\displaystyle \\lambda ^{*}(n)=\\tan \\left\\{{\\frac {1}{2}}\\arctan[g_{n}^{-12}]\\right\\}={\\sqrt {g_{n}^{24}+1}}-g_{n}^{12}}

>>Other appearances

>>>Little Picard theorem

The lambda function is used in the original proof of the `F33f`_`[Little Picard theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Little_Picard_theorem]`_`f, that an `F33f`_`[entire`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Entire_function]`_`f non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.`:cite-ref-18[`F5bf`_`[15`#cite-note-18]`_`f] Suppose if possible that `*f`* is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function `*z`* → ω(`*f`*(`*z`*)). By the `F33f`_`[Monodromy theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monodromy_theorem]`_`f this is holomorphic and maps the complex plane `!C`! to the upper half plane. From this it is easy to construct a holomorphic function from `!C`! to the unit disc, which by `F33f`_`[Liouville's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouville's_theorem_(complex_analysis)]`_`f must be constant.`:cite-ref-19[`F5bf`_`[16`#cite-note-19]`_`f]

>>>Moonshine

The function τ τ ↦ ↦ 16 / λ λ ( 2 τ τ ) − − 8 {\\displaystyle \\tau \\mapsto 16/\\lambda (2\\tau )-8} is the normalized `F33f`_`[Hauptmodul`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hauptmodul]`_`f for the group Γ Γ 0 ( 4 ) {\\displaystyle \\Gamma _{0}(4)} , and its `*q`*-expansion q − − 1 + 20 q − − 62 q 3 + … … {\\displaystyle q^{-1}+20q-62q^{3}+\\dots } , `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A007248 where q = e 2 π π i τ τ {\\displaystyle q=e^{2\\pi i\\tau }} , is the graded character of any element in conjugacy class 4C of the `F33f`_`[monster group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monster_group]`_`f acting on the `F33f`_`[monster vertex algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monster_vertex_algebra]`_`f.

>>Footnotes

`:cite-note-c115-2`!1.`! `F0af`_`[↑`#cite-ref-c115-2-0]`_`f Chandrasekharan (1985) p.115
`:cite-note-c109-3`!2.`! `F0af`_`[↑`#cite-ref-c109-3-0]`_`f Chandrasekharan (1985) p.109
`:cite-note-c110-4`!3.`! `F0af`_`[↑`#cite-ref-c110-4-0]`_`f Chandrasekharan (1985) p.110
`:cite-note-c108-5`!4.`! `F0af`_`[↑`#cite-ref-c108-5-0]`_`f Chandrasekharan (1985) p.108
`:cite-note-c63-6`!5.`! `F0af`_`[↑`#cite-ref-c63-6-0]`_`f Chandrasekharan (1985) p.63
`:cite-note-c117-7`!6.`! `F0af`_`[↑`#cite-ref-c117-7-0]`_`f Chandrasekharan (1985) p.117
`:cite-note-8`!7.`! `F0af`_`[↑`#cite-ref-8]`_`f Rankin (1977) pp.226–228
`:cite-note-9`!8.`! `F0af`_`[↑`#cite-ref-9]`_`f `:citerefborweinborwein1987`aBorwein, Jonathan M.; Borwein, Peter B. (1987). `*Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity`* (First ed.). Wiley-Interscience. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-471-83138-7. p. 103–109, 134
`:cite-note-jacobi-11`!9.`! `F0af`_`[↑`#cite-ref-jacobi-11-0]`_`f `:citerefjacobi1829`a`F33f`_`[Jacobi, Carl Gustav Jacob`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Carl_Gustav_Jacob_Jacobi]`_`f (1829). `*Fundamenta nova theoriae functionum ellipticarum`* (in Latin). p. 42
`:cite-note-13`!10.`! `F0af`_`[↑`#cite-ref-13]`_`f `:citerefborweinborwein1987`aBorwein, Jonathan M.; Borwein, Peter B. (1987). `*Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity`* (First ed.). Wiley-Interscience. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-471-83138-7. p. 152
`:cite-note-14`!11.`! `F0af`_`[↑`#cite-ref-14]`_`f `:citerefchowlaselberg1949`aChowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". `*Proceedings of the National Academy of Sciences`*. `!35`! (7): 373. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1073/PNAS.35.7.371. `F33f`_`[PMC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PMC_(identifier)]`_`f 1063041. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 45071481.
`:cite-note-15`!12.`! `F0af`_`[↑`#cite-ref-15]`_`f `:citerefchowlaselberg`aChowla, S.; Selberg, A. "On Epstein's Zeta-Function". `*EuDML`*. pp. 86–110.
`:cite-note-16`!13.`! `F0af`_`[↑`#cite-ref-16]`_`f `:citerefberndtchanzhang1997`aBerndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations". `*Transactions of the American Mathematical Society`*. `!349`! (6): 2125–2173.
`:cite-note-17`!14.`! `F0af`_`[↑`#cite-ref-17]`_`f `:citerefeymardlafon1999`aEymard, Pierre; Lafon, Jean-Pierre (1999). `*Autour du nombre Pi`* (in French). HERMANN. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 2705614435. p. 240
`:cite-note-18`!15.`! `F0af`_`[↑`#cite-ref-18]`_`f Chandrasekharan (1985) p.121
`:cite-note-19`!16.`! `F0af`_`[↑`#cite-ref-19]`_`f Chandrasekharan (1985) p.118

>>References

>>>Notes

`:cite-note-1`!note 1.`! `F0af`_`[↑`#cite-ref-1]`_`f λ λ ( τ τ ) {\\displaystyle \\lambda (\\tau )} is not a `F33f`_`[modular function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modular_form]`_`f (per the Wikipedia definition), but every modular function is a `F33f`_`[rational function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_function]`_`f in λ λ ( τ τ ) {\\displaystyle \\lambda (\\tau )} . Some authors use a non-equivalent definition of "modular functions".
`:cite-note-10`!note 2.`! `F0af`_`[↑`#cite-ref-10]`_`f For any `F33f`_`[prime power`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_power]`_`f, we can iterate the modular equation of degree p {\\displaystyle p} . This process can be used to give algebraic values of λ λ ( n i ) {\\displaystyle \\lambda (ni)} for any n ∈ ∈ N . {\\displaystyle n\\in \\mathbb {N} .}
`:cite-note-12`!note 3.`! `F0af`_`[↑`#cite-ref-12]`_`f sl ⁡ ⁡ a ϖ ϖ {\\displaystyle \\operatorname {sl} a\\varpi } is algebraic for every a ∈ ∈ Q . {\\displaystyle a\\in \\mathbb {Q} .}

>>>Other

• `:citerefabramowitzstegun1972`a`F33f`_`[Abramowitz, Milton`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Milton_Abramowitz]`_`f; `F33f`_`[Stegun, Irene A.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Irene_Stegun]`_`f, eds. (1972), `*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables`*, New York: `F33f`_`[Dover Publications`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dover_Publications]`_`f, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-486-61272-0, `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 0543.33001
• `:citerefchandrasekharan1985`a`F33f`_`[Chandrasekharan, K.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=K._S._Chandrasekharan]`_`f (1985), `*Elliptic Functions`*, Grundlehren der mathematischen Wissenschaften, vol. 281, `F33f`_`[Springer-Verlag`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer-Verlag]`_`f, pp. 108–121, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-540-15295-4, `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 0575.33001
• `:citerefconwaynorton1979`a`F33f`_`[Conway, John Horton`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=John_Horton_Conway]`_`f; `F33f`_`[Norton, Simon`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Simon_P._Norton]`_`f (1979), "Monstrous moonshine", `*Bulletin of the London Mathematical Society`*, `!11`! (3): 308–339, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1112/blms/11.3.308, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0554399, `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 0424.20010
• `:citerefrankin1977`a`F33f`_`[Rankin, Robert A.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Robert_Alexander_Rankin]`_`f (1977), `*Modular Forms and Functions`*, `F33f`_`[Cambridge University Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cambridge_University_Press]`_`f, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-521-21212-X, `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 0376.10020
• `:citerefreinhardtwalker2010`aReinhardt, W. P.; Walker, P. L. (2010), "Elliptic Modular Function", in `F33f`_`[Olver, Frank W. J.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Frank_W._J._Olver]`_`f; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), `*`F33f`_`[NIST Handbook of Mathematical Functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Digital_Library_of_Mathematical_Functions]`_`f`*, Cambridge University Press, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-521-19225-5, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2723248.

• Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.

• Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.

• Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.

>>External links

• Modular lambda function at Fungrim

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