Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum
$$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{m\tau+n}$$.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.