Ok, so this is a rather loose question (but one that has been on my mind for some time) so I don't expect any rigorous (but hopefully serious and thoughtful) answers.
Think of an L-function attached to some motive (I refrain from giving a more detailed description of what this means; if in doubt think of Artin L-functions). Is there in any sense a ''space'' (vector space, scheme,...) of L-functions?
I know that this is not a very precise question, but my thoughts on this are not very precise either (more like a vague dream world), but I know that there are people out there on this forum who know a lot about L-functions. Hopefully some of these can make my dream world a little more real, or shatter it completely.
I should also say that the recent philosophical question modularity gave me the courage (!) to ask the present one.
Edit: I should perhaps stress that I'm not quite sure myself of what I mean but I'm hoping for experts to give their informed opinions of what could be true.
Edit: Ok, to accomodate Gerhard P's comment below in some small way let me give the following motivation.
Suppose given a Galois representation (over some fixed base). I'm particularly thinking of those rep's with finite image. To such a representation you can attach an L-function as the Artin L-function of the fixed field of the kernel of the rep. Suppose now that you deform the representation in "a moduli space" (or in some parameter space) of Galois representations. Is there a sense in which the L-function deforms with it to a family of L-functions in some "space" of L-functions?
This can (probably) be generalized to other motives.