  In this paper, first by employing inequalities derived from the Opial
inequality due to David Boyd with best constant, we will establish new
unconditional lower bounds for the gaps between the zeros of the Riemann zeta
function. Second, on the hypothesis that the moments of the Hardy Z-function
and its derivatives are correctly predicted, we establish some explicit
formulae for the lower bounds of the gaps between the zeros and use them to
establish some new conditional bounds. In particular it is proved that the
consecutive nontrivial zeros often differ by at least 6.1392 (conditionally)
times the average spacing. This value improves the value 4.71474396 that has
been derived in the literature.
