Gaia survey offers a most enormous data volume and exquisite data quality compared to any ground-based projects. However, magnitude-dependent systematic effects at the 10 mmag level or higher are shown in Gaia DR2. We use the spectroscopy-based stellar color regression method with ∼0.7 million common stars between Gaia and LAMOST to acquire color corrections in G − GRP and GBP − GRP for both DR2 and EDR3. A sub-millimagnitude precision is achieved. Based on a 0.2 million sample with well-calibrated photometric data and spectroscopic informations from the LAMOST survey, we estimate the binary fractions of the filed GK dwarf stars. The general binary fraction is 42% ±1%, 49% ± 2% for the thick disk, and 39% ± 2% for the thin disk. For both the thin and thick disk stars, the binary fractions decrease toward higher [Fe/H], [α/H], and [M/H]. However, the suppressing impact is more significant for the thin disk stars than those for the thick disk stars. For a given [Fe/H], a positive correlation between multiplicity and [alpha/Fe] is in the thin disk and disappear in the thick disk. We suppose it is related to the stellar age distribution. Comparing with the close binary fraction study, we conjecture that the correlation between [Fe/H] and multiplicity changes with the period. This is probably due to the different formation and evolution process of the close and wide binaries.