Summary: In this paper, we study a uniqueness question of meromorphic functions of certain differential polynomials that share a nonzero finite value or have the same fixed points with the same of \(L\)-functions. The results in this paper extend the corresponding results from J. Steuding [Value distribution of \(L\)-functions. Berlin: Springer (2007; Zbl 1130.11044), p. 152], B. Q. Li [Proc. Am. Math. Soc. 138, No. 6, 2071–2077 (2010; Zbl 1195.30041)], M.-L. Fang [Comput. Math. Appl. 44, No. 5–6, 823–831 (2002; Zbl 1035.30017)] and C.-C. Yang and X. Hua [Ann. Acad. Sci. Fenn., Math. 22, No. 2, 395–406 (1997; Zbl 0890.30019)].