I have 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 tried to transform the sine to exponential but whitout result. This series is quite similar to third theta Jacobi elliptic function.
For details check this link page $\sim $ 575