  The quantum Rabi model (QRM) is widely recognized as an important model in
quantum systems, particularly in quantum optics. The Hamiltonian
$H_{\text{Rabi}}$ is known to have a parity decomposition $H_{\text{Rabi}} =
H_{+} \oplus H_{-}$. In this paper, we give the explicit formulas for the
propagator of the Schr\"odinger equation (integral kernel of the time evolution
operator) for the Hamiltonian $H_{\text{Rabi}}$ and $H_{\pm}$ by the Wick
rotation (meromorphic continuation) of the corresponding heat kernels. In
addition, as in the case of the full Hamiltonian of the QRM, we show that for
the Hamiltonians $H_{\pm}$, the spectral determinant is, up to a non-vanishing
entire function, equal to the Braak $G$-function (for each parity) used to
prove the integrability of the QRM. To do this, we show the meromorphic
continuation of the spectral zeta function of the Hamiltonians $H_{\pm}$ and
give some of its basic properties.
