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The present paper studies Hecke rings derived by the automorphism groups ofcertain algebras $L_p$ over the ring of $p$-adic integers. Our previous workconsidered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3)over the ring of $p$-adic integers. Although this Hecke ring is noncommutative,we showed that a formal power series with coefficients in this Hecke ringsatisfies an identity similar to the rationality of the Hecke series for $GL_2$due to E.~Hecke. In the present paper, we establish an analogous result in thecase of the Heisenberg Lie algebra of dimension 5 over the ring of $p$-adicintegers. In this case, our identity is similar to the rationality of the Heckeseries for $GSp_4$, due to G.~Shimura.

A formal power series over a noncommutative Hecke ring and the rationality of the Hecke series for $GSp_4$