This question is related to the following formula for the Riemann Xi function $\xi(s)$ which I believe converges for $\Re(s)>0$ (see the two figures following the question below).
(1) $\,\xi(s)=X^{-s}\left(\frac{1}{2}+\sum\limits_{n=1}^N\left(e^{-\frac{\pi\,n^2}{X^2}}+\frac{\pi\,n^2\,(s-1)\,E_{-\frac{s}{2}}\left(\frac{n^2\,\pi}{X^2}\right)}{X^2}\right)\right),\,\Re(s)>0\land X\to\infty\land N\gg X$
Note the sum over $n$ in formula (1) above can be partially simplified as illustrated in formula (2) below where I believe $\underset{X\to\infty}{\text{lim}}\vartheta_3\left(0,e^{-\frac{\pi}{X^2}}\right)=X$ (see my related question).
(2) $\quad\sum\limits_{n=1}^{\infty}e^{-\frac{\pi\,n^2}{X^2}}=\frac{1}{2}\left(\vartheta_3\left(0,e^{-\frac{\pi}{X^2}}\right)-1\right)$
Question: Is there a closed form for the following expression (which is also related to the sum over $n$ in formula (1) above) for $0<\Re(s)$, $0<\Re(s)<1$, or even just $\Re(s)=\frac{1}{2}$?
(3) $\quad\sum\limits_{n=1}^\infty\,n^2\,E_{-\frac{s}{2}}\left(\frac{n^2\,\pi}{X^2}\right)$
The following two figures illustrate formula (1) above for $\xi(s)$ in orange overlaid on the reference function $\xi(s)$ in blue where formula (1) is evaluated at $X=200$ and $N=1000$ in both figures.
Figure (1): Illustration of Formula (1) for $\xi(s)$
Figure (2): Illustration of Formula (1) for $\xi(1/2+i\,t)$