TITLE: Double series problems
QUESTION [6 upvotes]: How to calculate$$\sum_{n=-\infty}^{\infty}{\sum_{m=-\infty}^{\infty}{\frac{\left(-1\right)^n}{\left(6m\right)^2+\left(6n+1\right)^2}}}.$$Follow this,we first get $$\sum\limits_{k =  - \infty }^\infty  {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {6k + 1} \right)}^2} + {l^2}}}}  = \frac{{{l^2}}}{{{l^4} + 1}} - \frac{1}{{{l^2} + 1}} - \frac{\pi }{{12l}}\left( {\frac{{\sinh \frac{{\pi l}}{6}}}{{\cosh \frac{{\pi l}}{6} + \frac{{\sqrt 3 }}{2}}} + \frac{{\sinh \frac{{\pi l}}{6}}}{{ - \cosh \frac{{\pi l}}{6} + \frac{{\sqrt 3 }}{2}}}} \right).$$But it seems difficult to continue!

REPLY [6 votes]: The paper Two-dimensional series evaluations via the elliptic
functions of Ramanujan and Jacobi deals exactly with double sums of this kind and shows how to evaluate them in terms of elliptic functions.
The book Lattice sums then and now gives alternative methods to calculate lattice sums.