  This study concerns (not necessarily commutative) Hecke rings associated with
certain algebras and describes a formal Dirichlet series with coefficients in
the Hecke rings, which can be used to generalize Shimura's series. Considering
the case of the Heisenberg Lie algebra, an analog of the identity for Shimura's
series derived employing the rationality theorem, presented by Hecke and
Tamagawa, is established. Moreover, this analog recovers the explicit formulae
for the isomorphic zeta function and pro-isomorphic zeta function of the
Heisenberg Lie algebra shown by Grunewald, Segal and Smith.
