I know that this is valid for every $s$ in the half plane $\{s:\operatorname{Re}(s)>0\}$, but does the formula extend along with the analytic continuation of the $\Gamma$ function?
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2Yessir, it sure does! – Jan 26 '18 at 23:19
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See (https://math.stackexchange.com/questions/159333/understanding-the-analytic-continuation-of-the-gamma-function) – Jean Marie Jan 26 '18 at 23:29
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If two analytic functions coincide on a set that has an accumulation point, then they're equal. – egreg Jan 26 '18 at 23:31