Proof or source for this Hurwitz Zeta function identity?

Question

I need a proof or source for this identity:

$ \zeta '\left(z,\frac{q}{2}\right)-2^z \zeta '(z,q)+\zeta '\left(z,\frac{q+1}{2}\right)=\zeta(z,q)2^{z}\ln 2$

Here the derivative means the derivative by the first variable.

Any similar identities, particular cases and generalizations are very much appreciated.

Answer 1

This identity follows from the following distribution relation on the Wikipedia page: $$\displaystyle\sum_{m=0}^{n-1}\zeta\left(z,a+\frac{m}{n}\right)=n^z\zeta\left(z,na\right).\tag{1}$$ It suffices to set therein $a=\frac{q}{2}$, $n=2$ and differentiate once with respect to $z$.