Is the prime zeta function value
$$ P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots $$
a transcendental number ?
What about the following sum ? $$\sum_{p \in \mathrm{primes}} \frac{1}{p^p} = 0.2873582513062241797364180458789322069559088\ldots $$
For the last sum, Liouville's criterion might help because of the fast converging series.
The first sum might have been checked, as it is natural to come to such a sum. Perhaps, it is at least known if it is irrational.