The motivation to this question can be found in: About the number of zeros of the zeta function?
My question is: I am searching for similar results for Dirichlet series of the form:
$$∑_{n=1}^{∞}a_{n}/(n^{s})$$
Where can a rigorous formula of the number of zeros $N(T)$ be found?
You will want to suppose a few things about your Dirichlet series, as otherwise there is naught to say.
So suppose that your Dirichlet series is $$ L(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$ converges absolutely for $\mathrm{Re}(s) > 1$, satisfies a functional equation of the shape $\Lambda(s) := G(s) L(s) = \varepsilon \Lambda(1-s)$ where $G(s)$ is a product of gamma functions and $\pi$ and $\lvert \varepsilon \rvert = 1$, and has an Euler product $$ L(s) = \prod_{p} L_p(s).$$
Any function of this form also has zero-counting results similar to the Riemann zeta function. In fact, the analysis is very similar. It would be a good idea to go through a proof with the Riemann zeta on one side, and one of these generic $L$-functions on the other.
Good references for this are Montgomery and Vaughan's Analytic Number Theory (for an excellent beginning look) or Iwaniec and Kowalski (for a really, really high level reference look).
