Is this prime identity true : $\frac{27}{32} + \frac{\pi}{8}+\frac{5}{48 \pi} = 1 - \sum_{n=1}^{\infty} \frac{(-1)^n}{p_n}$?

Question

Consider the following alternating sum

$$P = 1 - \sum_{n=1}^{\infty} \frac{(-1)^n}{p_n}$$

where $p_n$ is the $n$ th prime.

Is it true that

$$ P = \frac{27}{32} + \frac{\pi}{8}+\frac{5}{48 \pi} $$

?

Answer 1

By $\text{A}078437$, $$ \zeta _p (1) - \left( \frac{27}{32} + \frac{\pi}{8} + \frac{5}{48\pi} \right) = - 9.53 \ldots \times 10^{ - 9} \neq 0. $$