Usually, \varpi is defined by the first equality below, but it has many equivalent forms: \begin{aligned} \varpi &= 2\int_0^1\frac{dt}{\sqrt{1-t^4}} = \sqrt2\int_0^\infty\frac{dt}{\sqrt{1+t^4}} = \int_0^1\frac{dt}{\sqrt{t-t^3}} = \int_1^\infty \frac{dt}{\sqrt{t^3-t}}\\[6mu] &= 4\int_0^\infty\Bigl(\sqrt[4]{1+t^{4}}-t\Bigr)dt = 2\sqrt2\int_0^1 \sqrt[4]{1-t^{4}}\mathop{dt} =3\int_0^1 \sqrt{1-t^4}dt\\[2mu] &= 2K(i) = \tfrac{1}{2}\Beta\bigl( \tfrac14, \tfrac12\bigr) = \tfrac{1}{2\sqrt2}\Beta\bigl( \tfrac14, \tfrac14\bigr) = \frac{\Gamma (1/4)^2}{2\sqrt{2\pi}} = \frac{2-\sqrt2}{4}\frac{\zeta(3/4)^2}{\zeta(1/4)^2}\\[5mu] &= 2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots, \end{aligned} where is the
complete elliptic integral of the first kind with modulus , is the
beta function, is the
gamma function and is the
Riemann zeta function. The lemniscate constant can also be computed by the
arithmetic–geometric mean M, \varpi=\frac{\pi}{M\bigl(1,\sqrt2\bigr)}. Gauss's constant is typically defined as the
reciprocal of the
arithmetic–geometric mean of 1 and the
square root of 2, after his calculation of M\bigl(1, \sqrt2\bigr) published in 1800:G = \frac{1}{M\bigl(1, \sqrt2\bigr)}John Todd's lemniscate constants may be given in terms of the
beta function B: \begin{aligned} A &= \frac\varpi2 = \tfrac14 \Beta \bigl(\tfrac14,\tfrac12\bigr), \\[3mu] B &= \frac{\pi}{2\varpi} =\tfrac14\Beta \bigl(\tfrac12,\tfrac34\bigr). \end{aligned}
As a special value of L-functions \beta'(0)=\log\frac{\varpi}{\sqrt{\pi}} which is analogous to \zeta'(0)=\log\frac{1}{\sqrt{2\pi}} where \beta is the
Dirichlet beta function and \zeta is the
Riemann zeta function. Analogously to the
Leibniz formula for π, \beta (1)=\sum_{n=1}^\infty \frac{\chi (n)}{n}=\frac{\pi}{4}, we have L(E,1)=\sum_{n=1}^\infty \frac{\nu (n)}{n}=\frac{\varpi}{4} where L is the
L-function of the
elliptic curve E:\, y^2=x^3-x over \mathbb{Q}; this means that \nu is the
multiplicative function given by \nu (p^n)=\begin{cases} p - \mathcal{N}_p, & p\in\mathbb{P}, \, n=1 \\[5mu] 0, & p=2,\, n\ge 2 \\[5mu] \nu (p)\nu (p^{n-1})-p\nu (p^{n-2}), & p\in\mathbb{P}\setminus\{2\},\, n\ge 2 \end{cases} where \mathcal{N}_p is the number of solutions of the congruence a^3-a\equiv b^2 \,(\operatorname{mod}p),\quad p\in\mathbb{P} in variables a,b that are non-negative integers (\mathbb{P} is the set of all primes). Equivalently, \nu is given by F(\tau)=\eta (4\tau)^2\eta (8\tau)^2=\sum_{n=1}^\infty \nu (n) q^n,\quad q=e^{2\pi i\tau} where \tau\in\mathbb{C} such that \operatorname{\Im}\tau >0 and \eta is the
eta function. The above result can be equivalently written as \sum_{n=1}^\infty \frac{\nu (n)}{n}e^{-2\pi n/\sqrt{32}}=\frac{\varpi}{8} (the number 32 is the
conductor of E) and also tells us that the
BSD conjecture is true for the above E. The first few values of \nu are given by the following table; if 1\le n\le 113 such that n doesn't appear in the table, then \nu (n)=0: \begin{array}{r|r|r|r} n & \nu (n) & n & \nu (n) \\ \hline 1 & 1 & 53 & 14 \\ 5 & -2 & 61 & -10 \\ 9 & -3 & 65 & -12 \\ 13 & 6 & 73 & -6 \\ 17 & 2 & 81 & 9 \\ 25 & -1 & 85 & -4 \\ 29 & -10 & 89 & 10\\ 37 & -2 & 97 & 18 \\ 41 & 10 & 101 & -2 \\ 45 & 6 & 109 & 6 \\ 49 & -7 & 113 & -14 \\ \end{array}
As a special value of other functions Let \Delta be the minimal weight level 1 new form. Then \Delta (i)=\frac{1}{64}\left(\frac{\varpi}{\pi}\right)^{12}. The q-coefficient of \Delta is the
Ramanujan tau function. == Series ==