Analytic continuation of $p(s)=\sum_{p\text{ prime}} e^{-p^s}$

Question

What is the analytic continuation of $p(s)=\sum_{p\text{ prime}} e^{-p^s}=e^{-2^s}+e^{-3^s}+e^{-5^s}+\cdots$ ?

I've attempted the case where the index of the sum is not prime, but is $n=1,2,3,\dots$. In this case we can use the Cahen-Mellin integral to get an analytic continuation.

However this technique doesn't work for $p(s)$.

$p(s)$ converges for real variable $s>0$.