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    J-function

    Result

    1728 J(τ)

    Series expansion at τ = 0

    196884 e^(-(2 i π)/τ) + e^((2 i π)/τ) + 21493760 e^(-(4 i π)/τ) + 864299970 e^(-(6 i π)/τ) + 20245856256 e^(-(8 i π)/τ) + 333202640600 e^(-(10 i π)/τ) + 4252023300096 e^(-(12 i π)/τ) + 744

    Derivative

    d/dτ(12^3 J(τ)) = -(1024 i (λ(τ) - 2) (λ(τ) + 1) (2 λ(τ) - 1) ((λ(τ) - 1) λ(τ) + 1)^2 K(λ(τ))^2)/(π (λ(τ) - 1)^2 λ(τ)^2)

    Alternative representation

    12^3 J(τ) = (4 12^3 (1 - λ(τ) + λ(τ)^2)^3)/(27 (1 - λ(τ))^2 λ(τ)^2)

    12^3 J(τ) = (12^3 (η(τ)^24 + 256 η(2 τ)^24)^3)/(1728 η(τ)^48 η(2 τ)^24)

    12^3 J(τ) = (12^3 (η(τ)^12 + 27 η(3 τ)^12) (η(τ)^12 + 243 η(3 τ)^12)^3)/(1728 η(τ)^36 η(3 τ)^12)

    Series representation

    12^3 J(τ) = 744 + e^(-2 i π τ) + sum_(k=1)^∞ e^(2 i k π τ) a_k for (a_k = (2 π sum_(j=1)^∞ (I_1((4 sqrt(k) π)/j) A_j(k))/j)/sqrt(k) and A_j(k) = sum_(h=0)^(-1 + j) e^(-(2 i π (h k + H(j, h)))/j) δ_(1, gcd(h, j)) for (h H(j, h)) mod j = -1)

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