I have several questions about the Eisenstein series: $G_k(\tau)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\frac{1}{(m+n\tau)^k}$ where {m,n} does not equal {0,0} and k is even and bigger that two.
I am aware of the formula $G_{4k}(i)=\frac{(2\omega)^{4k}}{(4k)!}H_{4k}$ by Hurwitz, but I don't know how to show it nor generate the Hurwitz numbers. All I know is that he arrives at the formula using the Weierstrass eliptic function.
Is there any other closed-form expression for the Eisenstein series? Are any of the zeroes known? Are there any closed expressions for the derivatives and integrals of the Eisenstein series?