formula of our zeta functions, we prove Theorem \[thm:main\]. Finally in § \[sec:localzeta424\], we will give the the details of the unramified computation and show the explicit formula of the zeta functions.

**Acknowledgments.** The authors thank Wen-Wei Li for helpful discussions and sharing his impeccably written preprints with us. Needless to say, this paper owes its inspiration to a series of his recent pioneering works. The authors also would like to thank Kimball Martin for pointing out that the difference between our result and Theorem \[thm:FH\] is reduced to a local problem. A further thank you to both Atsushi Ichino and Tamotsu Ikeda for useful comments and kindly answering our questions. The first author would like to thank Hang Xue for helpful comments. The second author thanks Akihiko Yukie for helpful discussions.

Preliminaries
=============

The space of pairs of quaternion algebras {#sec:space}
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Throughout this subsection, let $F$ be a field of characteristic not two. Let $\bar{F}$ denote an algebraic closure of $F$, and set $\underline{G}:=\GL_2(\bar{F})^3$, and $\underline{V}:=\bar{F}^2\otimes \bar{F}^2\otimes\bar{F}^2$. A rational irreducible representation $\underline{\rho}$ is defined by $v\cdot\underline{\rho}(g)=(v_1g_1)\otimes(v_2g_2)\otimes(v_3g_3)$ where $g=(g_1,g_2,g_3)\in \underline{G}$ and $v=(v_1,v_2,v_3)\in \underline{V}$. Then, the triple $(\underline{G},\underline{\rho},\underline{V})$ is a prehomogeneous vector space, that is, it has an open dense 