This paper is a continuation of a previous paper by the authors and S. Wakatsuki [Am. J. Math. 131, No. 6, 1525–1541 (2009; Zbl 1250.11038)] in which relations among Dirichlet series whose coefficients are class numbers of binary cubic forms are studied. The group \(\mathrm{SL}_2(\mathbb Z)\) acts on the space \(V_{\mathbb Q}\) of binary cubic forms over the rational number field by linear change of variables. In the previous paper [loc. cit.] the authors determined all of the \(\mathrm{SL}_2(\mathbb Z)\)-invariant lattices of \(V_{\mathbb Q}\) and studied properties of the zeta functions associated to these lattices. There are 10 such lattices, up to \(\mathbb Q^{\times}\)-multiplication, labelled as \(L_i\) and \(L_i^*\) for \(1\leq i \leq 5\). Following T. Shintani [J. Math. Soc. Japan 24, 132–188 (1972; Zbl 0223.10032)], for each \(1\leq i \leq 5\), there are zeta functions \(\xi_{ij}(s)\) and \(\xi_{ij}^*(s)\), \(1\leq j \leq 2\), associated to \(L_i\) and \(L_i^*\), respectively. For \(1\leq i \leq 3\), there are identities relating the corresponding zeta functions. Writing \[ \xi_i(s)=\begin{pmatrix} \xi_{i1}(s)\\ \xi_{i2}(s) \end{pmatrix}, \quad \xi_i^*(s)=\begin{pmatrix} \xi_{i1}^*(s)\\ \xi_{i2}^*(s) \end{pmatrix}, \text{ and } A=\left(\begin{matrix} 0 & 1 \\ 3 & 0 \end{matrix} \right), \] these identities have the simple expression \(\xi_i^*(s)=A\xi_i(s)\). For \(i=1\), this identity was conjectured by Y. Ohno [Am. J. Math. 119, No. 5, 1083–1094 (1997; Zbl 0893.11035)] and proved by J. Nakagawa [Invent. Math. 134, No. 1, 101–138 (1998; Zbl 1016.11014)] utilizing class field theory. The identities for \(i=2,3\) were derived by Ohno, Taniguchi and Wakatsuki [loc. cit.] by reducing to Nakagawa’s theorem for the case \(i=1\). For \(i=4,5\), \(\xi_i^*(s)\) and \(A\xi_i(s)\) no longer coincide. However, in the main theorem of the present paper, the authors prove that an identity of the same type holds for certain linear combinations of the zeta functions. The proof of this result is again accomplished by reducing to the case \(i=1\), now by looking at certain subsets of \(L_1\) which are invariant under congruence subgroups such as \(\Gamma_0(2)\) and \(\Gamma (2)\) and studying them by induction in the category of \(G\)-sets. The result is used to further study functional equations of the zeta functions and properties of related Dirichlet series.