  The zeta function attached to a finite complex $X_\Gamma$ arising from the
Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed
form expression was obtained by a combinatorial argument. This identity can be
rephrased using operators on vertices, edges, and directed chambers of
$X_\Gamma$. In this paper we reprove the zeta identity from a different aspect
by analyzing the eigenvalues of these operators using representation theory. As
a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of
the eigenvalues of the operators on vertices, edges, and directed chambers,
respectively.
