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The braid group on strands plays a central role in knot theory and low dimensional topology. 3-braids were classified, up to conjugacy, into normal forms. Basing on Burau's representation of the braid group, Birman introduced a simple way to calculate the Jones polynomial of closed 3-braids. We use Birman's formula to study the structure of the Jones polynomial of links of braid index 3. More precisely, we show that in many cases the normal form of the 3-braid is determined by the Jones polynomial and the signature of its closure. In particular we show that alternating pretzel links , which are known to have braid index 3, cannot be represented by alternating 3-braids. Also we give some applications to the study of symmetries of 3-braid links. [ABSTRACT FROM AUTHOR]