What is the state of the art for numerically computing the prime zeta function? There are several papers on the subject for the Riemann zeta function, such as this paper. This paper on the subject dates to 1968. Is there more recent work?
I am especially interested in the region near the natural boundary.
Wouldn't claim "the state of the art", but there's a simple idea (present in some form in High precision computation of Hardy-Littlewood constants by H. Cohen): for $\Re s>0$ and $s$ not a singularity, $$P(s)=\sum_{p\leqslant N}p^{-s}+\sum_{n\geqslant 1}\frac{\mu(n)}{n}\log\zeta_{p>N}(ns),$$ where we set $\zeta_{p>N}(s)=\zeta(s)\prod_{p\leqslant N}(1-p^{-s})$ and take $N$ suitably large (to make the last sum above converge quickly: for $s=\sigma+i\tau$ we have $\log\zeta_{p>N}(ns)\in\mathcal{O}(N^{-n\sigma})$ as $n\to\infty$).