I'm trying to show that the following sum converges to $0$ over the lattice $L = \mathbb{Z}[i]$ of Gaussian integers:
$$ 140\times\sum_{\substack{l \in L \\ l\neq 0}} l^{-6} = 0. $$
I don't really know how to attack this, I haven't studied complex analysis yet. What I've noticed is that the sums over the pure real and imaginary parts cancel out as $i^{-6} = -1$. So, that leaves all the terms not on the imaginary or real axis to deal with.
Would anyone have hints on how to proceed?