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What are standard books where one can lookup values of functions that cannot be calculated analytically?

For example, I want to know the value of $$S(\alpha)=\sum_{x=1}^\infty\sum_{y=-\infty}^\infty \frac{1}{\sqrt{x^2+y^2}^{~\alpha}}~,$$

but I don't know of an analytical solution. I can calculate the sums brute-force for any given value of $\alpha$, but to me it looks like something that many people may have calculated before.

Where would one look to find such values?

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A standard reference work for tables of mathematcal function values is Abramowitz and Stegun.

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