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How to compute the following integral $I_{\alpha,\beta}$
We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.)
$$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...
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