The Riemann zeta function and its various generalizations have been extensively studied by mathematicians worldwide. The L-functions are Selberg class functions with Riemann zeta function as the prototype and since L-functions are analytically continued as meromorphic functions, it is convenient to study the value distribution and uniqueness problems on L-functions and arbitrary meromorphic functions. Further, the fact that L-functions neither have a pole nor zero at the origin, but is having only possible pole at s = 1 helps us to study some of the classical results of Boussaf et al. [3] in terms of an L-function and an arbitrary meromorphic function. In this paper, by using the concept of weighted sharing and least multiplicity, we study the value distribution of an L-function and an arbitrary meromorphic function when certain type of differential polynomials generated by them share a non-zero small function with finite weight. Our results extends and improves some of the classical results due to Boussaf et al. (Indagationes Mathematicae 24(1):15-41, 2013).