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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 $\tau$.

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    $\begingroup$ For fixed $\tau$ in the upper half-plane, the function $s\to E_s(\tau)$ is also called an Epstein zeta function, 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, 2016 at 22:33
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    $\begingroup$ ... and the idea of "periods of Eisenstein series" is a well-established thing, going back to Hecke and Maass, and more examples in recent years. Erez Lapid and various of his collaborators have done many examples. Rankin-Selberg integral expressions for various L-functions can be considered examples of such periods. In this vein, Deligne's Corvallis (1977, AMS 1979) piece about special values is relevant. But/and what is the question? $\endgroup$
    – paul garrett
    Jul 28, 2016 at 22:52
  • $\begingroup$ Ok, that's interesting, thanks. But nevertheless, my question still isn't answered. I would like to know a list of known special values of $E(\tau, 0)$ for certain $\tau$. Or any special values whatsoever, for arbitrary $s$. You have talked about linear combinations, not values at specific points. $\endgroup$
    – Bruce Bartlett
    Jul 29, 2016 at 7:15
  • $\begingroup$ To give one example, the residues of the Eisenstein series appear in the computation of the fundamental domain of $G/\Gamma$ done by Langlands. $\endgroup$
    – Asaf
    Jul 29, 2016 at 7:56
  • $\begingroup$ The more natural objects are suitable linear combinations (over Heegner points) giving special values of ideal-class group L-functions, from which the individual values can be recovered by linear algebra. Values at $s=0$ are residues at $s=1$, which are non-zero only for the trivial ideal-class character. For general points $\tau$, but at $s=0$ or $s=1$, over $\mathbb Q$ we have the Kronecker limit formula. $\endgroup$
    – paul garrett
    Jul 29, 2016 at 12:46

1 Answer 1

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Combining paul garret's comments (see also his wonderful notes, Standard compact periods for Eisenstein series) with the class number formula, the functional equation of the Dedekind zeta function and the fact that $\zeta(0)=-1/2$, we get:

$$E(\tau,0)=\frac{\zeta^*_K(0)}{\zeta(0)}=\frac{2h_KR_K}{w_K}$$

where $h_K$ is the class number, $R_K$ the regulator and $w_K$ the order of the group of roots of unity of $K=\mathbb{Q(\tau)}$.

From here it should be very tractable computation for any reasonable $\tau$.

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