On a product formula of bivariate $p$-adic Gamma values

We establish a derivative formula of $p$-adic Shintani $L$-functions, thus those of totally real $p$-adic Hecke $L$-functions with trivial moduli. As an application, we present a product formula of bivariate $p$-adic Gamma values by specializing to genus $L$-functions, which takes a particularly simple form when the field is $\mathbf{Q}(\sqrt{3})$. In the appendix we explain how the product formula is supposed to encode a higher analytic analogue of the Davenport-Hasse relation.