I am trying to understand why the following sum converges $$\sum_{\lambda \in \Lambda\backslash\{0\}}\frac{1}{|\lambda|^3},$$ where $\Lambda=\{m+n\tau \mid m,n\in \mathbb{Z} \}$ with $\tau \in \mathbb{C},\text{Im}(\tau)>0$. I understand why the sum $$\sum_{m,n\in \mathbb{Z}^2\backslash\{(0,0)\}}\frac{1}{(m^2+n^2)^{3/2}}$$ converges (this is the case $\tau=i$), but I am having trouble generalizing this for arbitrary $\tau$.
Any help would be greatly appreciated.