In this chapter we shall collect some basic facts about L-functions derived from cusp-forms over the full modular group Г by way of the Hecke correspondence. We shall see in the next chapter that these L-functions appear as components of our explicit formula for the fourth power moment of the Riemann zeta-function. This fact is in no sense superficial: In the course of the proof of the mean value we shall have an expression involving Kloosterman sums, which is spectrally decomposed by means of Theorems 2.3 and 2.5. But this will be made initially only in a domain of relevant parameters which does not contain the point we are most interested in. Thus we shall face the problem of analytic continuation; and its solution will depend indispensably on the analytical properties of these L-functions. In this process of analytic continuation the multiplicative property of Hecke operators will play an important rôle. Hence we shall develop the essentials of Hecke's idea in the first section. Once the analytic continuation is completed and the explicit formula for the mean value is established, we shall need good spectro statistical estimates of special values of L-functions to extract quantitative information from the formula. Having this application in mind we shall give an account of spectral mean values of Hecke L-functions in the later sections of this chapter.