$$\Phi(s) = \frac{\Gamma(s)}{2^s-1} \quad(-1<s<0), \quad a_k = \frac{2\pi k}{\ln 2}i \quad(k\neq 0) $$
Now that $a_k$ are pure imaginary poles, I wanna get
$$\lim_{s\to a_k}(s-a_k)\Phi(s)$$
Using l'Hopital, $$\lim_{s\to a_k} \frac{\Gamma(a_k) +(s-a_k)\Gamma'(s)}{\ln 2}$$
Does $\Gamma'(a_k)$ converge so that the value is $\frac{\Gamma(a_k)}{\ln 2}$?
How to prove $\Gamma'(a_k) < \infty$ ??