  By introducing a new coordinate system, we prove that there are abundant new
periodic orbits near relative equilibrium solutions of the N-body problem. We
consider only Lagrange relative equilibrium of the three-body problem and
Euler-Moulton relative equilibrium of the N-body problem, although we believe
that there are similar results for general relative equilibrium solutions. All
of these periodic orbits lie on a 2d-dimensional central manifold of the planar
N-body problem. Besides d one parameter family of periodic orbits which are
well known as Lyapunov's orbits or Weinstein's orbits, we further prove that
periodic orbits are unexpectedly abundant: generically the relative measure of
the closure of the set of periodic orbits near relative equilibrium solutions
on the 2d-dimensional central manifold is close to 1. These abundant periodic
orbits are named Conley-Zender's orbits, since to find them is based on an
extended result of Conley and Zender on the local existence result for periodic
orbits near an elliptic equilibrium point of a Hamiltonian. In particular, the
results provide some evidences to support the well known claim of Poincar\'{e}
on the conjecture of periodic orbits of the N-body problem.
