Let $P(s)$ be the Prime zeta function.

Numerical evidence suggests these identities:

$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad (1)$$

$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(nk)}{k}=\log{\bigg(\frac{1}{a(n)}\frac{\pi^{2n}}{\zeta(n)}\bigg)}\qquad (2)$$

for natural $n$, where $a(n)$ is OEIS A002432 Denominators of $~\dfrac{\zeta(2n)}{\pi^{2n}}$.

In $a(n)$ we have $\zeta(2n)$ and in (2) $\zeta(n)$.

Is (1) and/or (2) true ?