  Comparison between the exact value of the spectral zeta function,
$Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results
of numeric and WKB calculations supports the conjecture by Bessis that all the
eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional
Schr\"odinger operators with complex potentials having a monotonic imaginary
part, the eigenfunctions (and the imaginary parts of their logarithmic
derivatives) have no real zeros.
