We study the L-values of an elliptic curve twisted by an Artin representation. Specifically, we consider the case in which the representation factors through a false Tate curve extension of Q. First, we consider a semistable elliptic curve E; we construct an integral-valued p-adic measure which interpolates the values the L-values of an Artin twist of E, at a family of finite-order character twists. To do this, we exploit the fact that such an L-value may be written as the Rankin convolution of two Hilbert modular forms, when the representation factors through the false Tate curve extension. Recent developments in non-abelian Iwasawa theory predict certain strong congruences between these p-adic L-functions, and we shall establish weakened versions of these congruences. Next, we prove analogous results for an elliptic curve with complex multiplication; we do this using work of Hida and Tilouine on the p-adic interpolation of Hecke L-functions over a CM-field. We go on to investigate the ratio of the automorphic and motivic periods associated to E in this setting. We describe how the p-valuation of this ratio may be explicitly calculated, and use the computer package MAGMA to produce some numerical examples. We end by proving a formula for the growth of this quantity in terms of the Iwasawa invariants associated to the two-variable extension of the CM-field.