Do L-functions exist for Half-integral weight modular forms?

You can certainly attach $L$-functions to half-integer weight eigenforms, but you don't get anything really new by doing so: they turn out be versions of $L$-functions of integer weight modular forms. More specifically, there is the "Shimura lifting" map from weight $k + 1/2$ to weight $2k$, which sends eigenforms to eigenforms; and the L-function of a half-integer weight eigenform will be closely related to that of its image under the Shimura lift. See e.g. here:

www.mathcs.emory.edu/~ono/REUs/archive/results/reu06shimura.pdf

In fact this turns out to be a very powerful way of studying the L-functions of integer weight forms (used, for instance, in Tunnell's work on the congruent number problem, which uses modular forms of weight 3/2 to understand the values at $s=1$ of the $L$-functions of twists of elliptic curves.

Upon David Loeffler's request, here is a more fleshed out version of my former comments:

In his comment, Nick Ramsey mentioned that the natural L-function for a half-integral weight modular form is really the entire family of quadratic twists of an L-function coming from a Shimura-type lift. I agree with Nick's perspective and it motivated me to work towards a more general framework. I don't mean this post as shameless self-promotion, but here's my preprint on Split Metaplectic Groups and their L-groups. I think this will probably not be widely read due to the excessive use of Hopf algebras and reliance on Lusztig's canonical bases. Fortunately, I've recently worked out ways to avoid these completely, and I will hopefully have another preprint up soon.

Back to the question at hand: the general non-metaplectic perspective is that an L-function can be produced from a pair $(\pi, \rho)$ where $\pi$ is an automorphic representation of $G_{\mathbb A}$ and $\rho$ is an algebraic representation of the L-group ${}^L G$. For classical modular forms, one might take $G = PGL_2$ and ${}^L G = SL_2(C) \times \Gamma$ where $\Gamma$ is the absolute Galois group of ${\mathbb Q}$.

My perspective on the metaplectic groups is that again, an L-function should be associated to a pair $(\pi, \rho)$ where $\pi$ is a genuine automorphic representation of $\tilde G_{\mathbb A}$ (this makes sense in a framework of Brylinski-Deligne, for example) and $\rho$ is an algebraic representation of a putative L-group ${}^L \tilde G$. In the simplest case, $\tilde G = Mp_2$ is the metaplectic group. My preprint is devoted to the construction of such an L-group.

The key subtlety, observed by Nick and others, is the ambiguity if one uses the Shimura correspondence as guidance. Indeed, work of Shimura and Waldspurger requires the choice of an additive character, and thus for the definition of an L-function. In my construction, this choice gets wrapped up in the choice of algebraic representation $\rho$ of the L-group.

Roughly speaking, the L-group ${}^L Mp_2$ of $Mp_2$ is noncanonically isomorphic to $SL_2(C) \times \Gamma$. It arises in my preprint from a somewhat magical/contrived twisting of both multiplication and comultiplication in the Hopf algebra of the direct product. A less contrived non-Hopfy approach will appear in a new paper sometime soon (I hope). I hope to tackle global issues as well.

It turns out that every nontrivial additive character $\psi$ of ${\mathbb A} / {\mathbb Q}$ can be used to generate an isomorphism from the L-group ${}^L Mp_2$ to the direct product $SL_2(C) \times \Gamma$, whence a natural 2-dimensional representation $\rho_\psi$. The L-functions produced by various choices of additive character (various isomorphisms of the L-group to the L-group of $PGL_2$) should comprise an orbit, under quadratic twisting, of a single L-function.

To summarize, the L-functions could be written $L(\pi, \rho_\psi)$ for $\pi$ a genuine automorphic representation of $Mp_{2n}$ and $\rho_\psi$ a two-dimensional representation of ${}^L Mp_{2n}$ coming from an additive character $\psi$.

Since the original question asked only for analytic continuation, functional equation, I'd like to add that the Mellin transform $\sum a_n n^{-s}$ of the half integral weight modular form $\sum a_n \exp(2 \pi i n z)$ has these two properties, but it lacks an Euler product decomposition even if the modular form is a Hecke eigenform, since the Fourier coefficients at square free indices are not multiplicatively related. A similar phenomenon occurs for Siegel modular forms with the so called Koecher-Maaß series; here the Fourier coefficients at matrices representing maximal lattices (Kern- resp. Stammformen in Brandt's terminology) have no multiplicative relation.