Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the compatible system of $\ell$-adic representations $H^1_\ell(E) \otimes \rho$.
Let $p$ be a prime where $E$ has good supersingular reduction. Is it known (maybe just in some specific examples) whether there exists a $p$-adic $L$-function interpolating the values $L(E, \rho \otimes \chi, 1)$, where $\rho$ is fixed but $\chi$ varies over all Dirichlet characters of $p$-power conductor? I gather this is known in some cases when $E$ has good ordinary reduction at $p$, but I'm specifically interested in the supersingular case.
EDIT: Since this question seems to have come alive again after over a year's inactivity, I will add a little clarification. What I'm hoping for are distributions of finite order on the cyclotomic Galois group $\Gamma \cong \mathbb{Z}_p^\times$ satisfying some reasonable interpolation formula linking their values at finite-order characters to the $L$-values I mentioned above; and I expect that one will have to make a choice of eigenvalue of crystalline Frobenius of E at p satisfying some small slope condition, as one does in the Amice-Velu-Vishik construction for $\rho = 1$. I fully expect that there will be some story involving decomposing these potentially unbounded distributions in terms of auxiliary bounded distributions as in Pollack's $\pm$-construction, but I'm not asking about that here.
