Cycles, L-functions and triple products of elliptic curves

A variant of a conjecture of Beilinson and Bloch relates the rank of the Griffiths group of a smooth projective variety over a number field to the order of vanishing of an L-function at the center of the critical strip. Presently, there is little evidence to support the conjecture, especially when the L-function vanishes to order greater than 1. We study 1-cycles on E^3 for various elliptic curves E/Q. In each of the 76 cases considered we find that the empirical order of vanishing of the L-function is at least as large as our best lower bound on the rank of the Griffiths group. In 11 cases this lower bound is two.