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In this paper, we study, both theoretically and experimentally, the threshold characteristics of a distributed feedback fiber laser that depend on the value of a phase shift introduced into the fiber Bragg grating structure. We show that as the phase shift possesses a noticeable birefringence, the laser oscillates at any phase shift value. We also reveal that the laser threshold is different for the cavity eigen polarizations and depends on the phase shift value. We derive a simple analytical formula to calculate the laser threshold in the case of pi phase shift; this formula can be utilized to estimate a minimal threshold value for the laser with certain active fiber and Bragg grating parameters. The developed theory allows us to fairly model the experimentally measured dependence of the laser threshold on induced phase shift value.