Behavior of the $L$-functions of cusp forms at $s=1$

Let $f(z)$ be a Hecke eigenform in the space $S_{2k}(\Gamma)$ of holomorphic $\Gamma$-cusp forms of even weight $2k$, $\Gamma=\mathrm{SL}(2,\mathbb Z)$; let $L_f(s)$ be the $L$-function of $f(z)$. The goal of this paper is to obtain some results on $L_f(1)$ as $k$ increases. In particular, we prove an analogue of the classical Landau theorem in the theory of Dirichlet $L$-functions and (under a very plausible hypothesis) an analogue of the famous Siegel theorem. Bibliography: 15 titles.