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I am interested in finding the exact value of the following sum.

$\sum_{l=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{\left(l^{2}+m^{2}\right)^{3/2}}.$

This looks similar to the Riemann zeta function (https://en.wikipedia.org/wiki/Riemann_zeta_function). I was wondering whether it is possible to evaluate the sum exactly.

Boa_Constrictor
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  • Unfortunately there is no consensus as to whether or not $0$ should be in $\Bbb N$, so I suggest you use other notation. – jjagmath Jul 12 '21 at 15:08
  • A function similar to the $\zeta$ function would have a complex variable, say $s$ instead of $3/2$. Did you try CAS already? – Raphael J.F. Berger Jul 12 '21 at 15:10
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    See this answer: https://math.stackexchange.com/questions/197496/series-involving-catalan-and-z – Sungjin Kim Jul 12 '21 at 15:36
  • The answer itself from @SungjinKim's comment is more helpful in this context than the question: https://math.stackexchange.com/a/198288/16078 – Greg Martin Jul 12 '21 at 17:27

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