Estimates for truncated prime zeta function

Question

We know that the prime zeta function $P(s) = \sum \limits_{p} p^{-s}$ is absolutely convergent for $\Re(s) > 1$ but I could not find an estimate for these values. More specifically, I am interested in the sum $\sum \limits_{x \le p \le y} p^{-s}$ when $s$ is a positive integer $\ge 2$. If such estimates are not available, even the numerical values of $P(s)$ for the above values of $s$ would suffice.