We present the Lyα luminosity functions (LFs) at z = 5.7 and 6.6 derived from a new large sample of 1266 Lyα emitters (LAEs) identified in total areas of 14 and 21 deg², respectively, based on the early narrowband data of the Subaru/Hyper Suprime-Cam survey. Together with careful Monte Carlo simulations that account for the incompleteness of the LAE selection and the flux estimate systematics in the narrowband imaging, we have determined the Lyα LFs with unprecedentedly small statistical and systematic uncertainties in a wide Lyα luminosity range of 10^42.8–43.8 erg s⁻¹. We obtain best-fit Schechter parameters of , $\phi ^{*}_{\mathrm{Ly}\alpha } = 0.85^{+1.87}_{-0.77} \ (0.47^{+1.44}_{-0.44}) \times 10^{-4}\:\mathrm{Mpc}^{-3}$, and at z = 5.7 (6.6). We confirm that our best-estimate Lyα LFs are consistent with the majority of the previous studies, but find that our Lyα LFs do not agree with the high number densities of LAEs recently claimed by Matthee/Santos et al.’s studies that may overcorrect the incompleteness and the flux systematics. Our Lyα LFs at z = 5.7 and 6.6 show an indication that the faint-end slope is very steep (α ≃ −2.5), although it is also possible that the bright-end LF results are enhanced by systematic effects such as the contribution from AGNs, blended merging galaxies, and/or large ionized bubbles around bright LAEs. Comparing our Lyα LF measurements with four independent reionization models, we estimate the neutral hydrogen fraction of the intergalactic medium to be $x_\mathrm{H\,{\small I}} = 0.3 \pm 0.2$ at z = 6.6, which is consistent with the small Thomson scattering optical depth obtained by Planck 2016.