I am currently working on some numerical code to evaluate the Hurwitz zeta function of complex arguments. My approach starts by using DLMF §25.11.4 and §25.11.3 to ensure that $\Re(a) > 0$:
$$ \zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}} $$
Then, I use DLMF §25.11.25 where I had broken down the integral into two parts and substituted $x^{-1}$ in the second integral to then compute these using a finite Tanh-Sinh integrator. For my application, this approach covers the $\Re(s) > 1$ branch entirely with satisfactory precision:
$$ \zeta\left(s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\,\mathrm{d}x $$
I am however unsure how to proceed when $\Re(s) \le 1$. Having done some tests on DLMF §25.11.9, I have arrived at the conclusion that it does not appear to converge for $a \in \mathbb{C}$ (DLMF claims that this holds for $\Re(s)>0$ if $0<a<1$; $\Re(s)>1$ if $a=1$):
$$ \zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}} \sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right) $$
However, even if there was a way for this to converge to an useful value, I would need to bring the value of $a$ down further if required and handle the cases of $a = 0$ and $a = 1$ somehow. How can I accomplish the goal stated in the title of the question?