The Burghelea-Friedlander-Kappeler (BFK)-gluing formula for the regularized zeta-determinants of Laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta-determinants of a one-parameter family of the Dirichlet-to-Neumann operators. When the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface, this constant is well known. In this paper, we discuss this constant in two cases: one is when a warped product metric is given near a cutting hypersurface, and the other is when a manifold is a product manifold. Especially in the first case, we use the result of Fucci and Kirsten [Commun. Math. Phys. 317, 635-665 (2013)] in which the regularized zeta-determinant of the Laplacian defined on a warped product manifold is computed.
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