Lyman-alpha luminosity function (LF) based on deep observations of four lensing clusters. The precise aim of the present study is to further constrain the abundance of Lyman-alpha emitters (LAEs) by taking advantage of the magnification provided by lensing clusters. We blindly selected a sample of 156 LAEs, with redshifts between 2.9<z<6.7 and magnification-corrected Lyman alpha luminosities in the range 39<LogL_Ly{alpha}_[erg/s]<43. The price to pay to benefit from magnification is a reduction of the effective volume of the survey, together with a more complex analysis procedure. To properly take into account the individual differences in detection conditions (including lensing configurations, spatial and spectral morphologies) when computing the LF, a new method based on the 1/Vmax approach was implemented. The LAE LF has been obtained in four different redshift bins with constraints down to logL_Ly{alpha}_=40.5. From our data only, no significant evolution of LF mean slope can be found. When performing a Schechter analysis including data from the literature to complete the present sample a steep faint-end slope was measured varying from {alpha}=-1.69^+0.08^_-0.08_ to {alpha}=-1.87^+0.12^_-0.12_ between the lowest and the highest redshift bins. The contribution of the LAE population to the star formation rate density at z~6 is ~<50% depending on the luminosity limit considered, which is of the same order as the Lyman-break galaxy (LBG) contribution. The evolution of the LAE contribution with redshift depends on the assumed escape fraction of Lyman-alpha photons, and appears to slightly increase with increasing redshift when this fraction is conservatively set to one.