For real $s>0$, let
$$S(s):=\sum\_{n=-\infty}^\infty e^{-n^2/(2s^2)}
=\vartheta \_3\left(0,e^{-1/(2 s^2)}\right),$$
where $\vartheta$ is the elliptic theta function.


Plotting suggests that the identity
\begin{equation}
 S(s)=s\sqrt{2\pi} 
\end{equation}
is true at least for $s\ge3/2$. Is it indeed?


This conjecture, with a plot, appeared as a part of [this answer](https://mathoverflow.net/a/397437/36721), but seems to warrant separate posting. Mathematica cannot prove or disprove this identity.



An identity for the elliptic theta function
To give an answer, adding to my comments, your formula doesn’t hold true, although the error is exponentially small as $s\to \infty$, as can be seen by Poisson summing, which transforms your sum to
$$\sqrt{2\pi}s\left(1+\mathcal{O}(e^{-2\pi^2 s^2})\right).$$


