We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be palindromic and to have non-negative integer coefficients with a combinatorial interpretation. Along the way, we uncover unexpected symmetries between the values of the zeta function at negative and positive integers, expressed at the level of their generating functions. Using these symmetries, we ultimately establish a functional equation of the type \( s \longleftrightarrow 1-s \) for a natural completion of the zeta function.