In this paper, the downlink spectral efficiency of multi-cell multi-user large-scale distributed antenna systems (DASs) is studied in the presence of pilot contamination. Based on the properties of Gamma distribution, the closed-form expression is derived for the downlink achievable rate with maximum ratio transmission (MRT). The ultimate rate is also given when the ratio of the total number of base station (BS) antennas to the number of users goes to infinity. Finally, the results are validated via numerical simulations. It is shown that the closed-form expression is very accurate, and the downlink achievable rate of large-scale DAS is much larger than that of co-located massive multiple-input multiple-output (MIMO) with the same antenna configuration.