Partial sum estimate of sum of reciprocals of prime powers

Question

I want to estimate the following sum $$ \sum_{p\leq x}\frac{1}{{p^n}}\tag{1} $$

Here n=2,3...

We already know by Mertens theorem:

$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k) \tag{2}$$

Here $E(k)=o(1)$

B is constant which has an explicit expression

I'm wondering if the sum (1) has similar expression as sum (2) particularly with the explicitly expressable constant ?