Value for sum of inverse of Gaussian integers

Question

So I was studying some stuff about lattices and at some point I reached the Eisenstein series of weight 2k given by $$G_{2k}(\Lambda)=\sum_{w\in\Lambda-\{0\}}w^{-2k}$$ Then I tried to think about "easy" cases of such series so I thought "let's see what happens when I take $\Lambda=\mathbb{Z}[i]$ over $\mathbb{C}$ and $k=1$ but couldn't really get a nice value for it, or there is no such nice value?

Thanks

Answer 1

I just found something useful on Complex Analysis, Stein & Shakarchi, chapter 9. They give an equivalent formula for the general case, so in this particular one $$G_2(\mathbb{Z}[i])=\frac{\pi^2}{3}-8\pi^2\sum_{r=1}^{\infty}\sigma(r)e^{-2\pi r}$$ where $\sigma(r)$ is the sum of the divisors of r, which is good for now.