It has been conjectured that for ε≥0 the entire spectrum of the non-Hermitian PT-symmetric Hamiltonian H_N = p² + x²(ix)^ε, where N = 2 + ε, is real. Strong evidence for this conjecture for the special case N = 3 was provided in a recent paper by Mezincescu (Mezincescu G A 2000 J. Phys. A: Math. Gen. 33 4911) in which the spectral zeta function Z₃(1) for the Hamiltonian H₃ = p² + ix³ was calculated exactly. Here, the calculation of Mezincescu is generalized from the special case N = 3 to the region of all N≥2 (ε≥0) and the exact spectral zeta function Z_N(1) for H_N is obtained. Using Z_N(1) it is shown that to extremely high precision (about three parts in 10¹⁸) the spectrum of H_N for other values of N such as N = 4 is entirely real.