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Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$

As from the title, I am currently dealing with this sum $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$ in particular with $p=1/2,3/2,...$ (but once solved for $p=1/2$ one can derive wrt $a$ and find the ot …
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