Zeta functions of complexes arising from PGL(3)

In this paper we obtain a closed form expression of the zeta function Z( _XΓ , u) of a finite quotient _XΓ of the Bruhat-Tits building of _PGL3 over a nonarchimedean local field F by a discrete cocompact torsion-free subgroup Γ of _PGL3 . Analogous to a graph zeta function, Z( _XΓ , u) is a rational function with two different expressions and it satisfies the Riemann hypothesis if and only if _XΓ is a Ramanujan complex.