Recently, the deviation of the ratios $R(D)$, $R(D^{*})$ and $R(J/psi)$ have been found between experimental data and the Standard Model predictions, which may be the hint of New Physics. In this work, we calculate these ratios within the Standard Model by using the improved instantaneous Bethe-Salpeter method. The emphasis is pad to the relativistic correction of the form factors. The results are $R(D)=0.312 ^{+0.006}_{-0.007}$, $R(D^*)= 0.249^{+0.001}_{-0.002}$, $R(D_s)=0.320 ^{+0.009}_{-0.009}$, $R(D^*_s)=0.251 ^{+0.002}_{-0.003}$, $R(eta_c)=0.384 ^{+0.032}_{-0.042}$, and $R(J/psi)=0.267 ^{+0.009}_{-0.011}$, which are consistent with predictions of other models and the experimental data. The semileptonic decay rates and corresponding form factors at zero recoil are also given.
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