Given $\tau$ in the upper half plane, define the normalized real-analytic Eisenstein series by $$ E(\tau, s) = \frac{1}{2} \sum_{(m,n)}' \frac{y^s}{|m\tau + n|^{2s}} $$ It is initially defined for $\text{Re} (s) > 1$, but then analytically continued to the whole plane, except for a simple pole at $s=1$.

What are some known special values of this function? The only known values I have seen on the internet are $E(i, s) = 4 \zeta(s) \beta(s)$. I am especially interested in $E(\tau, 0)$ for any other values of $\tau$.

Epsteinzetafunction, easily google-able. The sum of these values over $\tau$ running through Heegner points for a given negative fundamental discriminant produces $\zeta_k(s)/\zeta(2s)$, where $k$ is the corresponding complex quadratic field extension of $\mathbb Q$. This was known to Hecke. Other linear combinations of special values at Heegner points produce ideal-class characters of those fields. Values at "generic" $\tau$ seem to be much less well-understood. $\endgroup$ – paul garrett Jul 28 '16 at 22:335more comments