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 $\tau$. 



Special values of real analytic Eisenstein series
Combining paul garret's comments (see also his wonderful notes, [Standard compact periods for Eisenstein series](http://www.math.umn.edu/~garrett/m/v/eis_std_periods.pdf)) 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 $\tau$.


