  We prove algebraicity of critical values of certain Spin $L$-functions. More
precisely, our results concern $L(s, \pi \otimes \chi, \mathrm{Spin})$ for
cuspidal automorphic representations $\pi$ associated to a holomorphic Siegel
eigenform on $\mathrm{GSp}_6$, Dirichlet characters $\chi$, and critical points
$s$ to the right of the center of symmetry. We use the strategy of relating the
$L$-values to properties of Eisenstein series, and a significant portion of the
paper concerns the Fourier coefficients of these Eisenstein series. Unlike in
prior algebraicity results following this strategy, our Eisenstein series are
on a group $G$ that has no known moduli problem, and the $L$-functions are
related to the Eisenstein series through a non-unique model.
