  We explore the effect of zeros at the central point on nearby zeros of
elliptic curve L-functions, especially for one-parameter families of rank r
over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's
Specialization Theorem, for t sufficiently large the L-function of each curve
E_t in the family has r zeros (called the family zeros) at the central point.
We observe experimentally a repulsion of the zeros near the central point, and
the repulsion increases with r. There is greater repulsion in the subset of
curves of rank r+2 than in the subset of curves of rank r in a rank r family.
For curves with comparable conductors, the behavior of rank 2 curves in a rank
0 one-parameter family over Q is statistically different from that of rank 2
curves from a rank 2 family. Unlike excess rank calculations, the repulsion
decreases markedly as the conductors increase, and we conjecture that the r
family zeros do not repel in the limit. Finally, the differences between
adjacent normalized zeros near the central point are statistically independent
of the repulsion, family rank and rank of the curves in the subset.
Specifically, the differences between adjacent normalized zeros are
statistically equal for all curves investigated with rank 0, 2 or 4 and
comparable conductors from one-parameter families of rank 0 or 2 over Q.
