I'm reading about the Weierstrass zeta function. In this context,
$\phi(z)=\zeta(z)-\pi\bar{z}$
is periodic over the lattice
$$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$
If we take $w\in\mathcal{L}\setminus\{\mathbf{0}\}$, then
$$\phi(z+w)-\phi(z)=w\sum\_{q\in\mathcal{L}\setminus\{0\}}q^{-2}-\pi\bar{w}.$$ Since $\phi$ is periodic, we get
$$\frac{1}{\pi}\sum\_{q\in\mathcal{L}\setminus\{0\}}q^{-2}=\frac{\bar{w}}{w}.$$
At this point, I wonder the value of this summation. If someone knows about this, please let me know.


**Question.** What is the value of
$$\sum\_{q\in\mathcal{L}\setminus\{0\}}q^{-2}?$$



The sum of $q^{-2}$ over nonzero Gaussian integers
The series
$$\sum\_{q\in\mathcal{L}\setminus\{0\}}q^{-2}$$
is not absolutely convergent. If we arrange the terms in groups of four of shape $\{\pm q,\pm iq\}$, then the series converges to zero since
$$q^{-2}+(iq)^{-2}+(-q)^{-2}+(-iq)^{-2}=0,\qquad q\in\mathcal{L}\setminus\{0\}.$$
For more general sums of similar shape, see [Eisenstein series](https://en.wikipedia.org/wiki/Eisenstein_series).


