$L$-functions are certain meromorphic functions generalizing the Riemann zeta function. They are typically defined by what is called an $L$-series which is then meromorphically extended to the complex plane.
Many L-functions have mutually similar deep features like satisfaction of a functional equation etc. The generalized Riemann conjecture is concerned with zeros of the Dirichlet zeta function? for which the L-series is complicated from the classical Riemann case by the presence of the additional parameter, the Dirichlet character.
wikipedia L-function, Dirichlet L-function, special values of L-function, generalized Riemann hypothesis, Artin L-function, equivariant L-function, functional equation), modularity theorem
Terrence Tao’s blog: Distinguished Lecture Series III: Shou-wu Zhang, “Triple L-series and effective Mordell conjecture”