The extra-zero conjecture for p-adic L-functions

Let $V$ be a $p$-adic representation satisfying conditions C1–4a, let $D$ be a regular subspace of $\mathbf D_{\mathrm{st}}(V)$, and let $e=\mathrm{rank}(\mathbf W_0)$. In the two cases a) and b) specified in the source, let $\ell(V,D)$ be the associated $\ell$-invariant. Extra-zero conjecture. The $p$-adic $L$-function $L_p(V,D,s)$ has a zero of order $e$ at $s=0$, and

$$\lim_{s\to 0}\frac{L_p(V,D,0)}{s^e}=\ell(V,D)\\,\mathcal E^+(V,D)\\,\Omega_p(M,D)\frac{L(M,0)}{\Omega_\infty(M)}.$$

Here $\mathcal E^+(V,D)$ is obtained from $\mathcal E(V,D)$ by excluding zero factors. The supplied status evidence says that the nonvanishing of $\ell(V,D)$ is a difficult open problem, so the conjectural formula remains open.