We consider the finite element approximation of fractional powers of regularly accretive operators via the DunfordTaylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito and J. E. Pasciak,
IMA J. Numer. Anal., 37(2016), No. 3, 12451273] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided.