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On a particular case of Dirichlet series [closed]

I've this series: $$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$ where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $. I need the limit of series like an analytic function of …