I have two questions regarding Non-holomorphic Eisenstein series:


$E\_s(\tau)=\sum\_{m,n}\frac{Im(\tau)^s}{|m+n \tau|^{2s}}$


where the sum runs over all the integers and we exclude $(0,0)$. The questions are:


1.- Does anyone know the expression for $E\_s(i)$? I feel this should be known.


2.- Are the $E\_s$ orthogonal with respect to some inner product? one problem seems to be that they don't decay at $\tau=i \infty$.


Thank you very much!



Non-holomorphic Eisenstein series
Hecke and Maass already knew that (your normalization of) $E\_s(i)$ is (4 times) the Dedekind zeta function of the Gaussian integers $\mathbb Z[i]$ evaluated at $s$.


You are correct that these Eisenstein series are not in $L^2$ for the $SL\_2(\mathbb R)$-invariant measure $dx\,dy/y^2$ descended to the (finite-volume) quotient $SL\_2(\mathbb Z)\backslash \mathfrak H$. Nevertheless, if their constant terms (0th Fourier coefficients) are truncated at any particular fixed height $y=a$, the resulting truncated Eisenstein series have inner products with each other given by the Maass-Selberg relations. 


Yes, the non-truncated Eisenstein series have differing eigenvalues for the invariant Laplacian, but, not being in $L^2$, there's no sensible "orthogonality" to be proven.


