Comprehensive low-temperature specific heat data C(T,H) of Na_0.35CoO2-1.3H_2O with temperature T down to 0.6 K and the magnetic field H up to 8 T are presented. For the normal state, the values of gamma_n=13.94 mJ/mol K2, and Debye temperature 362 K are reported. At zero field, a very sharp superconducting anomaly was observed at Tc=4.5 K with DeltaC/gamma_nTc=1.45. The superconducting volume fraction is estimated to be 47.4 % based on the consideration of entropy balance at Tc for the second-order superconducting phase transition. In the superconducting state, the electronic contribution C_es at H=0 can be well described by the model of the line nodal order parameter. In low H, gamma(H) H^1/2 which is also a manifestation of the line nodes. The behaviors of both Tc(H) and gamma(H) suggest the anisotropy of Hc2 or possible crossovers or transitions occurring in the mixed state.
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