Let E be a rational elliptic curve with CM by the ring of integers of a quadratic imaginary field, and let p be an odd prime of good ordinary reduction for E. The prime p splits in K. Then we have at least two essentially different ways to attach to E a p-adic avatar of its Hasse-Weil L-series (which we know it coincides with the Hecke L-series of its grossencharakter and that of its conjugate). In fact, working over Q, we have the p-adic L-function constructed by Mazur and Swinnerton-Dyer, and working over K, we can use Katz's work to attach to the grossencharakter of E an analogous one-variable p-adic L-function.

Do we know how these p-adic L-functions are related to each other?

(They are both conjecturelly related to the arithmetic of E via respective p-adic versions of the Birch and Swinnerton-Dyer conjecture, but I am wondering -- and this is the stress of the above question -- if the precise relation between the mere different constructions has been studied.)