Where you might want to start: The classical approach is based on special functions, and given e.g. here: http://www-math.mit.edu/~kedlaya/Math254B/zetafunction.pdf (I found this directly with google). I think the standard reference for such things is Neukirch "Algebraic Number Theory" and the later chapters on $L$ functions in this text.
A more elegant point of view: Tate's thesis gives the modern picture, but it is not free available, e.g. it is the last chapter in Cassels & Fr\"{o}hlich - Algberaic number theory. It is quiet self contained and very pleasant to read, if you know the basics about the Fourier transform of an locally compact abelian group. To learn the Fourier analysis, I recommend the first chapter of Rudin - Fourier analysis on groups as a start, and to translate every statement to the locally compact group $\mathbb{R}$ to get a good idea, what is going on. I think that Tate's approach is much more enlightening than the classical one, and there are many people which have rewritten parts of his thesis in various lecture notes, which are freely available online (use google). The key point of Tate's interpretation is that the class number formula is interpreted as a certain volume, and all classical functions, which turn up in the classical arguments, arise more naturally.
