Gross's conjecture on the leading term of the p-adic L-function

Let $F$ be a number field, let $H/F$ be a finite cyclic CM extension, let $p$ be a prime, and let $R$, $R_0$, and $R_1$ be the subsets of primes of $F$ above $p$ that split, ramify, and remain in the third case in $H$, respectively. Put $r_\chi=\#R$, let $\chi$ be the relevant character, let $L_p(\chi\omega,s)$ be the $p$-adic $L$-function, and let $\mathscr{R}_p(\chi)$ be the regulator of $p$-units of $H$. Gross's conjecture.

$$\operatorname{ord}_{s=0}L_p(\chi\omega,s)\geq r_\chi.$$

$$\frac{L_p^{(r_\chi)}(\chi,0)}{r_\chi!L(\chi,0)}=\mathscr{R}_p(\chi)\prod_{\mathfrak p\in R_0}(1-\chi(\mathfrak p)).$$

3. $\mathscr{R}_p(\chi)\neq0$, and hence $\operatorname{ord}_{s=0}L_p(\chi\omega,s)=r_\chi$. This is the $p$-adic analogue of the classical order-of-vanishing and leading-term formulas for the Artin $L$-function; the regulator nonvanishing is the substantive unresolved component in the formulation given here.