  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.
