Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$? Is there an upper bound for this count when we count zeros with $|\Im s|\leq T?$
In particular, what is known about zeta function? I know there are some explicit version of Von Mangoldt theorem that have been done for Zeta function. I want to know what are some relevant results for any $L(s,\chi).$
I guess there should be some results available for general $L$ functions, but I am unable to find them myself.
Any help would be appreciated. Thanks in advance.
Added: I meant an explicit formula for upper bound when I asked for one.
Let's state a general zero-counting form for a general $L$-function. Take an $L$-function with the functional equation
$$ \Lambda(s) := L(s) Q^s \prod_{\nu = 1}^N \Gamma( \alpha_\nu + \beta_\nu ) = \omega \overline{\Lambda(1 - \overline{s})}.$$
Define the degree of $L$ to be
$$ d_L := 2 \sum_{\nu} \alpha_\nu. $$
We also define the convenience variable
$$ \alpha := \prod_\nu \alpha_\nu^{2 \alpha_\nu}. $$
In practice, $d_L$ is an integer for natural $L$-functions. It is $1$ for $\zeta(s)$ and Dirichlet $L$-functions $L(s, \chi)$.
Then the number of nontrivial zeros of $L(s)$ of the form $\rho = \sigma + i t$ with $\lvert t \rvert < T$ is given by
$$ N(T) = \frac{d_L}{2 \pi} T \log \frac{4T}{e} + \frac{T}{2 \pi} \log(\alpha Q^2) + O(\log T). $$
We expect all of these to be on the $1/2$ line, but we don't know that. In many (maybe all?) cases we know at least that a positive proportion of the zeros are are the $1/2$ line. For degree $d_L \leq 2$, we also know that $100$ percent of zeros are within $\epsilon$ of the $1/2$ line for any $\epsilon$.
The $N(T)$ asymptotic is well-known and can be found (or quickly derived) from standard analytic number theory texts such as Montgomery and Vaughan, Iwaniec and Kowalski, or the classic books on $\zeta(s)$ by Heath-Brown or Titchmarsh.
Proving that there is a positive proportion of zeros on the line is also classic and also in Montgomery and Vaughan or Titchmarsh, at least for $\zeta(s)$. I don't know if anyone has written this down for $L(s, \chi)$, but it should be substantially similar.
Broadly, if you have questions about how someone proved something about $\zeta(s)$, it's a good idea to look through Titchmarsh's books on $\zeta$. There is an enormous wealth of techniques.