THE O(ALPHA-S3) CORRECTIONS TO SIGMA-TOT(E+E--] HADRONS) AND GAMMA(TAU- -]NU-TAU + HADRONS) IN QCD

We present the results of the total analytical reevaluation of the massless next-next-to-leading alpha-s3 QCD correction to R(S) = sigma-tot(e+e- --> gamma --> hadrons)/sigma(e+e- --> mu+ mu-). In numerical form the obtained result in the standard MSBAR-scheme for five active flavors is R(S) = 3/11[1 + alpha-s/pi + 1.409(alpha-s/pi)2 - 12.805(alpha-s/pi)3]. Including this correction in the fit of the experimental data in the energy range 7-57 GeV increases the value of LAMBDA-MSBAR about 30% and gives LAMBDA-MSBAR congruent-to 570(-320)+450 MeV. We also get the O(alpha-s3) approximation of R-tau = GAMMA(tau- --> v-tau + hadrons)/GAMMA(tau- --> v-tau-e-v(e)BAR) = 3[1 + alpha-s/pi + 5.2023(alpha-s/pi)2 + 26.366(alpha-s/pi)3]. Using the obtained results we determine the scheme invariants rho-2 for R(S) and R-tau. They turn out to be negative.