  Notable results on the special values of $L$-functions of Siegel modular
forms were obtained by J. Sturm in the case when the degree $n$ is even and the
weight $k$ is an integer. In this paper we extend this method to half-integer
weights $k$ and arbitrary degree $n$, determining the algebraic field in which
they lie. This method hinges on the Rankin-Selberg method; our extension of
this is aided by the theory of half-integral modular forms developed by G.
Shimura. In the second half, an analogue of P. B. Garrett's conjecture is
proved in this setting, a result that is of independent interest but that bears
direct applications to our first results. It determines exactly how the
decomposition of modular forms into cusp forms and Eisenstein series preserves
algebraicity and, ultimately, the full range of special values.
