The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $zeta$-functions to efficiently compute values of spectral $zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $tau$. Depending on the underlying boundary conditions, we express the $zeta$-function values in terms of a fundamental system of solutions of $tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $zeta$-function through a Liouville transformation and provide an explicit expression for the $zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schr{o}dinger operators with zero, piecewise constant, and a linear potential on a compact interval.
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