Value distribution of L-functions and uniqueness questions of F. Gross

We study the value distribution of L-functions and uniqueness questions of meromorphic functions sharing one or two sets with L-functions. The results in this paper extend the corresponding results from [J. Steuding, Value Distribution of L-Functions, Lect. Notes Math., Vol. 1877, Springer-Verlag, Berlin, Heidelberg, 2007, p. 152] and [B.Q. Li, A result on value distribution of L-functions, Proc. Am. Math. Soc., 138(6):2071-2077, 2010]. The studied questions in this paper are related to the questions of Gross from [F. Gross, Factorization of meromorphic functions and some open problems, in Complex Analysis, Kentucky 1976. Proceedings of the Conference Held at the University of Kentucky, May 18-22, 1976, Lect. Notes Math., Vol. 599, Springer-Verlag, Berlin, Heidelberg, 1977, pp. 51-69]. Here the meromorphic functions are of finitely many poles in the complex plane.