Greenberg's exceptional-zero conjecture for ordinary motives

Let $G(T,V)$ and $\mathcal{E}(V)$ be as above, and let $g$ be the number of vanishing factors of $\mathcal{E}(V)$. Let $\mathcal{E}^*(V)$ denote the product of the nonzero factors, and let $\mathcal{L}(V)$ be a nonzero error term. Greenberg's exceptional-zero conjecture. One has

$$G(T,V)=\mathcal{L}(V)\mathcal{E}^*(V)\frac{L(V,0)}{\Omega}\frac{T^g}{g!}+O(T^{g+1}).$$

This predicts that the order of the trivial zero equals the number of vanishing Euler factors and identifies the leading term in terms of a conjectural error term generalizing the Mazur–Tate–Teitelbaum phenomenon.