nt.number theory - Why do zeta functions contain so much information?

This question is a major open problem. One theoretical framework has been proposed that would explain why all the various zeta and L-functions (the ones attached to number fields, function fields, modular forms, algebraic varieties, Hecke characters, ...) have the same nice properties (analytic continuation, special values that "contain so much information," ...). It is known as the theory of motives. It is somewhat stymied by the fact that no one can (with proof) define the right category of motives, nor has anyone been able to for some 40 years. Milne's lecture notes "What is a motive?" are a great introduction to the theory; http://www.jmilne.org/math/xnotes/mot.html

For a particular answer to your question, the so-called Equivariant Tamagawa Number Conjecture in the theory of motives encompasses every one of the "information-packed" formulas we have about special values of L-functions. Of course, no one expects a proof any time soon (since such a proof would at once prove long outstanding questions about the category of motives, Birch and Swinnerton-Dyer's conjecture and the Stark conjectures).

Here's one more concrete idea. One important gadget in algebraic geometry is etale cohomology. Most L-functions that we know of can be defined in terms of the eigenvalues of various Frobenius maps (global and local) acting on etale cohomology groups. In many cases, by clever arguments using analogues of classical Betti cohomology theorems (Poincare duality, Lefschetz fixed point theorems, ...), we can express the results on special values of L-functions in terms of etale cohomology. In other words, the conceptual reason that L-functions defined from different objects behave so similarly to one another is that all these objects are algebro-geometric, and etale cohomology ought to be an awful lot like Betti cohomology. Of course, this is more believable for a variety over the complexes than it is for Spec Z, but it's a start.