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used only for functions based on gamma, not functions with some obscure relation to gamma

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"unexpected" residue formula for $\Gamma^3(s)/(\Gamma(3s)(e^{2\pi is}-1)) $

There is a related problem in my current work: to find the residue of the following function at any negative integer $s=-n$: $$f(s)=\frac{\Gamma^3(s)}{\Gamma(3s)(e^{2\pi is}-1)}$$ It seems to be a ted …
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