Sony A6000 has a Sony E lens mount and currently there are 102 native lenses available for this mount.
Sony A6000 doesn't have a sensor based image stabilization system but 23 of these lenses already comes with optical image stabilization.
Since Sony A6000 has a APS-C sensor, it has a focal length multiplier of 1.5x so you have to multiply lenses original focal length with this multiplier to find the Full Frame equivalent focal length of the lens when mounted on A6000.
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Sony A6000 Lenses with Image Stabilization
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