We discuss a specific class of regular-singular Laplace-type operators with matrix coefficients. Their zeta determinants were studied by Kirsten, Loya, and Park [Manuscr. Math. 125, 95 (2008)] on the basis of the Contour integral method, with general boundary conditions at the singularity and Dirichlet boundary conditions at the regular boundary. We complete the arguments of Kirsten, Loya, and Park by explicitly verifying that the Contour integral method indeed applies in the regular-singular setup. Further we extend the zeta-determinant computations to generalized Neumann boundary conditions at the regular boundary and apply our results to compute zeta determinants of Laplacians on a bounded generalized cone with relative ideal boundary conditions.
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