A question on something relating to analytic continuation of a $\Gamma()$-based function

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$$f(s) = \frac{\Gamma(s)}{2^s-1}, \space (-1<Re(s)<0 : fundamental \space strip)$$

There exsists $\gamma$ that $0<Re(s)\le \gamma$, such that $f(s) = O({|s|}^{-p}), (p>1)$

That is, How to evaluate the rightmost $\gamma$ that $|{\frac{\Gamma(s)}{2^s-1}}||s|^p \le C$ ?