We study the behavior of scale-free networks, having connectivity distribution P(k) approximately k(-lambda), close to the percolation threshold. We show that for networks with 3<lambda<4, known to undergo a transition at a finite threshold of dilution, the critical exponents are different than the expected mean-field values of regular percolation in infinite dimensions. Networks with 2<lambda<3 possess only a percolative phase. Nevertheless, we show that in this case percolation critical exponents are well defined, near the limit of extreme dilution (where all sites are removed), and that also then the exponents bear a strong lambda dependence. The regular mean-field values are recovered only for lambda>4.