Completion of local zeta functions associated with a certain class of homogeneous cones

Abstract

It is well known that the Riemann zeta function can be completed to the Riemann xi function $\xi(s)$ in the sense that its functional equation has a higher symmetric form $\xi(1-s)=\xi(s)$. In the previous paper (Tohoku Math. J. 72 (2020), 349--378), we give an explicit formula of functional equations between local and global zeta functions associated with a homogeneous cone and with its dual cone. In this paper, we consider a completion of these local zeta functions and show that, for a certain class of homogeneous cones, the associated local zeta functions admit a kind of completion forms.