I find a series in the paper of Borwein, it reads $$ \sum_{(x,y)\in Z^2/(0,0)}\frac{(-1)^{x+y}}{(x^2+3y^2)^s}=-2(1+2^{1-s})\alpha(s)L_{-3}(s) $$ where $\alpha(s)$ is alternating zeta function, and $L_{\pm d}(s)$ is the Dirichlet-$L$ function. How to prove it? I am not really familiar with the special tricks actually.

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    $\begingroup$ From the look of the paper this is a very non trivial result. It seems to follow as a particular case of a more general result, but this is also explained in the article. I might be wrong but I doubt anyone will be able to give any simpler answer (or even a different answer) than that provided. I would advice you to try to follow the paper and ask specific (smaller) doubts along the way. Good luck, that paper seems a tough task. $\endgroup$ – Rogelio Molina Jun 4 '15 at 6:37
  • $\begingroup$ See here for a similar question. You can't apply factorization directly here as unlike $\mathbb{Z}[i]$, the ring $\mathbb{Z}[\sqrt{-3}]$ is not a unique factorization domain, but the integral closure in $\mathbb{Q}(\sqrt{-3})$, ie the ring $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ is UFD. $\endgroup$ – r9m Jun 4 '15 at 9:07

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