calculus - sum of series of the form $sumlimits_{n=-infty}^{infty}-frac{(1+2n)}{2}exp(-frac{(1+2n)^2}{4})$.

Consider the series

$\sum\limits_{n=-\infty}^{\infty}-\frac{(1+2n)}{2}\exp(-\frac{(1+2n)^2}{4})$.

By using Mathematica I found the sum of the series is zero.

Question: I can Show that the series is convergent but I couldn't find its sum. Could anyone please help to find the sum of the series as $0$ ? Please help me, I have really no idea to find its sum.

Thank in advance.