We present an analytical calculation of the spectra of CMB anisotropies and polarizations generated by relic gravitational waves (RGWs). As a substantial extension to the previous studies, three new ingredients are included in this work. First, the analytic C TT l and C TE l are given; especially the latter can be useful to extract signal of RGWs from the observed data in the zero-multipole method. Second, a fitting formula of the decaying factor on small scales is given, coming from the visibility function around the photon decoupling. Third, the impacts by the neutrino free-streaming (NFS) is examined, a process that occurred in the early universe and leaves observable imprints on CMB via RGWs. It is found that the analytic C TT l and C TE l have profiles agreeing with the numeric ones, except that C TT l in a range l 10 and the first trough of C TE l around l $ 75 have some deviations. With the new damping factor, the analytic C EE l and C BB l match with the numeric ones with the maximum errors only $3% up to the first three peaks for l 600, improving the previous studies substantially. The correspondence of the positions of peaks of C XX l and those of RGWs are also demonstrated explicitly. We also find that NFS reduces the amplitudes of C XX l by ð20% $ 35%Þ for l ' ð100 $ 600Þ and shifts slightly their peaks to smaller angles. Detailed analyses show that the zero multipoles l 0 , where C TE l crosses 0, are shifted to larger values by NFS. This shifting effect is as important as those caused by different inflation models and different baryon fractions.