On the distribution of values of $L(1,\mathrm{sym}^2f)$

Let $S_k(\mathrm{SL}(2,\mathbb{Z}))^+$ be the set of holomorphic Hecke eigencuspforms $f$ of weight $k$ with respect to $\mathrm{SL}(2,\mathbb{Z})$. Let $L(s,\mathrm{sym}^2f)$ be the symmetric square of the Hecke L-function of a cusp form $f$. The moments of $L(1,\mathrm{sym}^2f)$, $f\in S_k(\mathrm{SL}(2,\mathbb{Z}))^+$, are computed for a pure imaginary order. The limiting distribution of $\log L(1,\mathrm{sym}^2 f)$, $f\in S_k(\mathrm{SL}(2,\mathbb{Z}))^+$, is studied in the weight aspect. Namely, the limiting distribution function, the limiting characteristic function and its Euler product are investigated, and the rate of convergence of frequencies to the limiting distribution is measured.

As a consequence, new facts on the limiting distribution of $\mathrm{SL}(1,\mathrm{sym}^2f)$ are obtained not only in the case of the holomorphic Hecke eigencuspforms $f$, but also in the case of the Hecke-Maass eigencuspforms $f$.