In this paper we investigate the distribution of zeros of the independence polynomial of hypergraphs of maximum degree Δ. For graphs the largest zero-free disk around zero was described by Shearer as having radius λs(Δ)=(Δ−1)Δ−1/ΔΔ. Recently it was shown by Galvin et al. that for hypergraphs the disk of radius λs(Δ+1) is zero-free; however, it was conjectured that the actual truth should be λs(Δ). We show that this is indeed the case. We also show that there exists an open region around the interval [0,(Δ−1)Δ−1/(Δ−2)Δ) that is zero-free for hypergraphs of maximum degree Δ, which extends the result of Peters and Regts from graphs to hypergraphs. Finally, we determine the radius of the largest zero-free disk for the family of bounded degree k-uniform linear hypertrees in terms of k and Δ.