Eisenstein Series on Siegel Space

I am going to assume you are asking about the (degenerate) Siegel Eisenstein series on $\mathrm{Sp}(2n)$. If that is the case, I think you should check out

Piatetski-Shapiro and Rallis, "Rankin triple products", section 4

and

Kudla and Rallis, "Poles of Eisenstein series and $L$-functions".

As far as I am aware, these are the canonical references for the harder analytic properties of the constant term and Fourier expansions of these Eisenstein series. But they are written in the adelic language, and offer perhaps a more in-depth treatment than you are looking for. So, you should be able to find an easier treatment just for the case of $\mathrm{Sp}(4)$, but I do not know a reference at the moment.