Actually this is a well known question. N. Lev and A. Olevskii have shown the following theorem:
Theorem (Lev, Olevskii) Given any 1 < p < 2 one can find two vectors in $l^1(Z)$, such that one is cyclic in $l^p(Z)$ and the other is not, but their Fourier transforms have an identical set of zeros.
The same result follows for $L^p(R)$.
Look here for example or on arxiv under Olevskii or Lev. This means more or less that for $p\neq 1,2$, there can be no characterization of $L^p$ generators in terms of the zero set of the Fourier transform. Hope this helps.