TITLE: Special values of real analytic Eisenstein series
QUESTION [5 upvotes]: 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 $\tau$.

REPLY [6 votes]: 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 $\tau$.