Let $f(s)$ be a somewhat zeta like function defined on the complex plane as :
$$f(s) = \sum_n p(n) n^{-s}= 1 + 2\cdot2^{-s}+ 3\cdot3^{-s} + 5\cdot4^{-s} + 7\cdot5^{-s} + \dots$$
where the coefficients $p(n)$ are the noncomposite numbers ( $1$ and the primes )
Consider solving $f(s)=0$.
It appears all but a finite amount of zero's have $\Re s\leq 2$.
Also it appears infinitely many zero's get arbitrary close to the line $\Re s=2$.
Is that true and how to prove or disprove those assumed properties ?