  The Rankin convolution type Dirichlet series $D_{F,G}(s)$ of Siegel modular
forms $F$ and $G$ of degree two, which was introduced by Kohnen and the second
author, is computed numerically for various $F$ and $G$. In particular, we
prove that the series $D_{F,G}(s)$, which share the same functional equation
and analytic behavior with the spinor $L$-functions of eigenforms of the same
weight are not linear combinations of those. In order to conduct these
experiments a numerical method to compute the Petersson scalar products of
Jacobi Forms is developed and discussed in detail.
