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The alloys of non-centrosymmetric superconductor, Re$_3$W, which werereported to have an $\alpha$-Mn structure [P. Greenfield and P. A. Beck, J.Metals, N. Y. \textbf{8}, 265 (1959)] with $T_\mathrm{c}=9 $K were prepared byarc melting. The ac susceptibility and low-temperature specific heat weremeasured on these alloys. It is found that there are two superconducting phasescoexisting in the samples with $T_\mathrm{c1}\sim9 $K and $T_\mathrm{c2}\sim7$K, both of which have a non-centrosymmetric structure as reported previously.By analyzing the specific heat data measured in various magnetic fields, wefound that the absence of the inversion symmetry does not lead to the deviationfrom a s-wave pairing symmetry in Re$_3$W.

$S$-wave pairing symmetry in non-centrosymmetric superconductor Re$_3$W