values of the fit parameters. The restricted data points are shown in Figure 1 by filled symbols. In the main analysis we use all the data points that satisfy the condition aQ$_{µ}$ < 4 π/ 16 in order to use as much information from the lattice data as possible with the uncorrelated fit. In particular, we can take a wider range of Q 2 with this choice, that improves the stability of the fit. In other words, with the limited data points ( Q$_{µ}$ ≤ 2 π/ 16) the χ 2 fit is sometimes trapped in a local minimum depending on the initial values for the fit parameters. We therefore decided to use the uncorrelated fit for the enlarged data points ( Q$_{µ}$ < 4 π/ 16) to obtain the fit parameters, and then to check the value of χ $^{2}$/ dof for the limited data points ( Q$_{µ}$ ≤ 2 π/ 16) taking account of the correlation. Since the value of χ $^{2}$/ dof is 1.7, we do not expect the bias due to this procedure larger than one standard deviation, assuming that the full correlated fit should give χ $^{2}$/ dof ∼ 1. Thus, we conservatively assign a systematic error ± 0 . 003 for α (5) s ( M$_{Z}$ ), which is equal to the size of the statistical error of one standard deviation. This procedure can be avoided if the lattice data are obtained at finer lattice spacings so that one can cover the same physical range of Q 2 with smaller lattice momenta. As described above, the discretization effect is significant in our lattice data especially when we try to cover large enough Q 2 range. We estimate the associated error using lattice perturbation theory. Since the discretization effect is most important in the large momentum region, the perturbation theory can be used to estimate its size. We calculate the one-loop diagram of VPF, Π PT V + $_{A}$( Q ), with local and conserved currents in lattice perturbation theory, and compare them with the continuum perturbation theory. This provides an estimate of the discretization effect at the zeroth order of α$_{s}$ . Because the discretization error itself is a small effect, its calculation at the leading order gives a reasonably precise estimate. Figure 5 shows the result as a function of ( aQ ) $^{2}$. Since the lattice regularization violates rotational invariance, the result is not a completely smooth function of ( aQ ) $^{2}$, as shown by squares in the plot, which correspond to representative points of ( aQ ) 2 in our lattice calculation. 13 B. Discretization effect