A numerical test of the generalized Birch and Swinnerton-Dyer conjecture

In its crudest form the generalized Birch and Swinnerton-Dyer relates the order of vanishing of an L-function at the center of the critical strip to the rank of a Chow group. This talk describes the conjecture and an attempt to put it to a modest test in the case of motives of the form Sym$^3H^1(E)$, where E is an elliptic curve over Q. This is joint work with Joe Buhler and Jaap Top.