  We study Ruelle's type zeta and $L$-functions for a torsion free abelian
group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the
imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8,
and in particular, such a zeta function has no determinant expression. Thus,
conversely, expressions like Euler's product for the determinant of the
Laplacians of the torus $\bR^{\n}/\G$ defined via zeta regularizations are
investigated. Also, the limit behavior of an arithmetic function arising from
the Ruelle type zeta function is observed.
