The Riemann zeta function ζ(s) was introduced by Euler and Riemann and is an important object in number theory. The special values of Riemann zeta function have striking arithmetic properties: ζ(k) is a rational number for every negative odd integer k, and Kummer congruence asserts that these values vary continuously under the p-adic topology. L-functions are natural generalizations of the Riemann zeta function. Starting from the Riemann zeta function, we will give some examples of L-functions, including Dirichlet L-functions and more general automorphic Lfunctions. Their rationality and p-adic continuity properties will be emphasized.