There is an article on Wikipedia about prime zeta function (PZF):

https://en.m.wikipedia.org/wiki/Prime_zeta_function

In that article , there is table of fairly accurate values of PZF for different $s$ .

We all know that

$\zeta(2n)=π^n\mathbb{Q}$.

So my question is :

Are there conjectured values of PZF such that they are the combination of well known transcendental numbers like $\pi$ and $e$ (like $\zeta(2n)$ above) and are very close to values given in the article ?