On the Distribution of Values of $L(1,f)$

Let $S_k(N)^+$ be the set of newforms of weight $k$ for $\Gamma_0(N)$, and let $L(s,f)$, $f\in S_k(N)^+$, be the Hecke $L$-function of the form $f$. It is proved that for every integer $m\ge1$, $k=2$ and $N=p\to\infty$
$$ \sum_{f\in S_2(N)^+}\,L^m(1,f)=\frac{1}{12}B_m N+O(N^{1-\alpha}), $$
where $B_m$ is a constant defined in the paper, and $\alpha=\alpha(m)>0$ is a certain constant. This result implies the existence of the distribution function of the sequence
$$ \{L(1,f),\,f\in S_2(N)^+\},\quad N=p\to\infty, $$
and also yields an explicit expression for the corresponding characteristic function.