Let L be a finite dimensional Lie algebra over a field k of characteristic zero and let S(L) be its symmetric algebra, equipped with its natural Poisson structure. First a sufficient condition is given for the Poisson semi-center Sz(S(L)) to be a polynomial algebra. It turns out that this condition holds for many nilpotent Lie algebras, in which case Sz(S(L)) coincides with the Poisson center of S(L). Then, using this and other methods, we are able to give an explicit description for the Poisson center for all complex, nilpotent Lie algebras of dimension at most seven. As a bonus we can produce in each case a maximal Poisson commutative subalgebra of S(L). Finally, all these results carry over to the enveloping algebra U(L) of L.