Geodesic
The distribution of the values of Hecke $L$-functions at~1
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 15-32

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Let $S_2(q)$ be the set of primitive forms in the space $S_2(\Gamma_0(q))$ of holomorpic $\Gamma_0(q)$-cusp forms of weight $2$. Let $f\in S_2(q)$ and let $L_f(S)$ be the $L$-function of $f(z)$. It is proved that the set $\{\log L_f(1)$ has a limit distribution function. The rate of convergence to this limit function is estimated. 
@article{ZNSL_2004_314_a1,
     author = {E. P. Golubeva},
     title = {The distribution of the values of {Hecke} $L$-functions at~1},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {15--32},
     publisher = {mathdoc},
     volume = {314},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a1/}
}
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E. P. Golubeva. The distribution of the values of Hecke $L$-functions at~1. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 15-32. http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a1/