Nancy Childress' book on class field theory has a nice chapter, even though it is more or less taken (in parts) from Washington's chapter 3 and 4.
Lang's "Algebraic number theory" also has a chapter I think.
Serre's "A course in arithmetic" is also nice.
Edit: there is a rather thorough introduction to L-functions in Volume 2 of Cohen's Number theory. I haven't seen it myself, but from the table of contents it seems very detailed and from scratch-approach.
As Kevin mentions Fröhlich-Taylor has a chapter on L-functions, but I find their book a little hard to read. But you might feel differently.
I'll report back if I think of anything else.