In this paper, we define fair computations in the pi-calculus. We follow Costa and Stirling's approach for CCS-like languages but exploit a more natural labeling method of process actions to filter out unfair process executions. The new labeling allows us to prove all the significant properties of the original one, such as unicity, persistence and disappearance of labels. It also turns out that the labeled pi-calculus is a conservative extension of the standard one. We contrast the existing fair testing with those that naturally arise by imposing weak and strong fairness as defined by Costa and Stirling. This comparison provides the expressiveness of the various fair testing-based semantics and emphasizes the discriminating power of the one already proposed in the literature.