A trace formula derivation of the functional equation for symmetric square L-functions

This paper revisits the functional equation of the symmetric-square $L$-function from the perspective of trace formulas. We show that, for level-one holomorphic cusp forms and even Hecke–Maass cusp forms, this symmetry can be recovered directly from the Petersson and Kuznetsov trace formulas. Although the functional equation itself is classical, our aim is to provide a concrete and transparent example of how the analytic structure of an $L$-function can emerge from a comparison of the spectral and geometric sides of a trace formula. The argument leads to an averaged reciprocity identity and treats the holomorphic and Maass settings within the same general framework. In this way, the paper offers a simple model for extending trace-formula methods beyond standard $L$-functions.