Ohno's conjecture on zeta functions of pairs of ternary quadratic forms

Let $L$ and $\hat{L}$ be the two lattices in the prehomogeneous vector space whose zeta functions are $\xi_i(L,s)$ and $\xi_i(\hat{L},s)$ for $i=1,2,3$, as defined in the paper. Ohno's conjecture. The zeta functions satisfy

$$\xi_1(\hat{L},s)=\xi_1(L,s)+\xi_3(L,s), \qquad \xi_2(\hat{L},s)=2\xi_2(L,s), \qquad \xi_3(\hat{L},s)=3\xi_1(L,s)-\xi_3(L,s).$$

Equivalently,

$$3\xi_1(\hat{L},s)+\xi_3(\hat{L},s)=2\left(3\xi_1(L,s)+\xi_3(L,s)\right),$$ $$\xi_2(\hat{L},s)=2\xi_2(L,s),$$ $$\xi_1(\hat{L},s)-\xi_3(\hat{L},s)=-2\left(\xi_1(L,s)-\xi_3(L,s)\right).$$

The conjecture relates the four Dirichlet series associated with the two lattices; it is motivated by the preceding coefficient identities and numerical examples. The paper states that the conjecture was proved by the author in 1998.