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, but seems to warrant separate posting. Mathematica cannot prove or disprove this identity.

`With[{s=2},N[EllipticTheta[3,0,E^(-1/(2s^2))]/(s Sqrt[2Pi]),50]]`

gives the answer`1.0000000000000000000000000000000001024500455847086`

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