  The main purpose of this note is to understand the arithmetic encoded in the
special value of the $p$-adic $L$-function
$\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ associated to a triple of
modular forms $(f,g,h)$ of weights $(2,1,1)$, in the case where the classical
$L$-function $L(f\otimes g\otimes h,s)$ - which typically has sign $+1$ - does
not vanish at its central critical point $s=1$. When $f$ corresponds to an
elliptic curve $E/\mathbb{Q}$ and the classical $L$-function vanishes, the
Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that
$\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ is either $0$ (when
the order of vanishing of the complex $L$-function is $>2$) or related to
logarithms of global points on $E$ and a certain Gross--Stark unit associated
to $g$. We complete the picture proposed by the Elliptic Stark Conjecture by
providing a formula for the value
$\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ in the case where
$L(f\otimes g\otimes h,1)\neq 0$.
