  In this article we obtain an explicit formula for certain Rankin-Selberg type
Dirichlet series associated to certain Siegel cusp forms of half-integral
weight. Here these Siegel cusp forms of half-integral weight are obtained from
the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama
correspondence. The integral weight version of the main theorem had been
obtained by Katsurada and Kawamura. The result of the integral weight case is a
product of $L$-function and Riemann zeta functions, while half-integral weight
case is a infinite summation over negative fundamental discriminants with
certain infinite products. To calculate explicit formula of such Rankin-Selberg
type Dirichlet series, we use a generalized Maass relation and adjoint maps of
index-shift maps of Jacobi forms.
