Apparently I have been exploring Riemann Zeta Function and have lately come across $L$-functions. After going through few papers, I realised that many mathematicians are giving their full time knowing the extreme values of $L$-functions. I need to know the following:
- What is the motivation to study these areas?
- What information do we get after finding central values of these functions?
Thank you in advance.
The OP probably meant the local extrema of the $\Bbb{R\to R}$ function $$\Xi(t)=(1/2+it)(1/2-it)\pi^{-(1/2+it)/2}\Gamma((1/2+it)/2)\zeta(1/2+it)$$ obtained from the functional equation.
Under the RH $$\Xi(t)=\Xi(0)\lim_{K\to \infty}\prod_{k=1}^K (1-\frac{t}{\Im(\rho_k)})^{e_k}(1+\frac{t}{\Im(\rho_k)})^{e_k}$$ where $\rho_k$ are the non-trivial zeros of $\zeta$ and $e_k$ their multiplicity.
$f_K(t)=\prod_{k=1}^K (1-\frac{t}{\Im(\rho_k)})^{e_k}(1+\frac{t}{\Im(\rho_k)})^{e_k}$ is a real polynomial whose all roots are real, its derivative $f_K'$ has at least one zero between each consecutive zero of $f_K$ (at the maximum of $|f_K|$ between those consecutive zeros) so we obtain $2K-1+\sum_{k=1}^K 2(e_k-1)$ zeros for $f_K'$, thus we have found them all.
Since the zeros of $f_K'$ converge locally uniformly to those of $\Xi'$ we know all the zeros of $\Xi'$,
and hence if the RH is true then there is exactly one local extremum of $\Xi$ between two consecutive zeros.
It works the same way for other L-functions.