Where are the solutions of the equations
$$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$
Since the functions on the LHS are the integral of the zeta function and the alternating zeta function, it might make a difference if one assumes RH or not.
I am considering zeroes in the entire complex plane , hence after analytic continuation. Although I am mainly interested in the zeross with $\operatorname{Re} s>-1$.
Is it true that all non real zeroes are in the critical strip $0<\operatorname{Re} s<1$ apart from at most a finite amount ?
I wondered if integral representations are insightful.
It is clear that for $\operatorname{Re} s>\alpha$ there is no solution when the positive real $\alpha$ is sufficiently large.
I assume these integrals of these zeta functions cannot be expressed as an infinite product over primes, is that true ?