This paper is devoted to the study of the lifespan of the solutions of the primitive equations for less regular initial data. We interpolate the globall well-posedness results for small initial data in ${\dot{H}}^{\frac{1}{2}}$ given by the Fujita-Kato theorem, and the result from [6] which gives global well-posedness if the Rossby parameter $\epsilon$ is small enough, and for regular initial data (oscillating part in ${\dot{H}}^{\frac{1}{2}}\cap {\dot{H}}^1$ and quasigeostrophic part in $H^1$).