Ramanujan-Eisenstein Series

Let $t$ be a discriminant,

$$q\equiv -e^{-\pi\sqrt{t}},$$ (1)

then

$$E_2(q) \equiv L(q)\equiv 1-24\sum_{k=1}^\infty {(2k+1)q^{2k+1}\over 1-q^{2k+1}}$$

$$= \left({2K\over \pi}\right)^2(1-2k^2)$$ (2)

$$E_4(q) \equiv M(q)\equiv 1+240\sum_{k=1}^\infty {k^3q^{2k}\over 1-q^{2k}}$$

$$= \left({2K\over \pi}\right)^4(1-k^2k'^2)$$ (3)

$$E_6(q) \equiv N(q)\equiv 1-504\sum_{k=1}^\infty {k^5q^{2k}\over 1-q^{2k}}$$

$$= \left({2K\over \pi}\right)^6(1-2k^2)(1+{\textstyle{1\over 2}}k^2k'^2).$$ (4)

See also Klein's Absolute Invariant, Pi

References

Borwein, J. M. and Borwein, P. B. ``Class Number Three Ramanujan Type Series for $1/\pi$.'' J. Comput. Appl. Math. 46, 281-290, 1993.

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.