  The contribution, E, of hyperbolic elements to the scalar Casimir energy on a
compact quotient of the upper half hyperbolic plane is computed for a
propagation operator conformal in three dimensions. Due to the proliferation of
prime closed geodesics, the series form for the Casimir energy has an IR
divergence. The expression for E is given as a sum of polylogarithms which
allows the divergence to be isolated and rendered finite by an {\it ad hoc}
Ramanujan renormalisation. The remaining part of E is computed using the
specific lower length spectrum of the (2,3,7) triangle and a universal
asymptotic form for larger lengths. The tentative value of E found is such as
to make the total conformal Casimir energy on the triangle probably negative.
