For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor).
Such conjectures represent formal statements concerning the philosophy that special values of L-functions contain arithmetic information.
More precisely, Lp(s, χ) is the unique continuous function of the p-adic number s such that for positive integers n divisible by p − 1.
The right hand side is just the usual Dirichlet L-function, except that the Euler factor at p is removed, otherwise it would not be p-adically continuous.
Deligne & Ribet (1980), building upon previous work of Serre (1973), constructed analytic p-adic L-functions for totally real fields.