An　r-uniform　hypergraph　is　linear　if　every　two　edges　intersect　in　at　most　one　vertex.　Given　a　family　of　r-uniform　hypergraphs　F,　the　linear　Turán　number　exrlin(n,F)　is　the　maximum　number　of　edges　of　a　linear　r-uniform　hypergraph　on　n　vertices　that　does　not　contain　any　member　of　F　as　a　subhypergraph.　For　each　k≥3,　the　linear　k-cycle　Ck　is　the　3-uniform　linear　hypergraph　with　edges　h1,…,hk　such　that　for　every　1≤i≤k−1,　|hi∩hi+1|=1,|hk∩h1|=1　and　hi∩hj=∅　for　all　other　pairs　{i,j},i≠j.　It　is　proved　by　Collier-Cartaino,　Graber,　Jiang　[3]　and　Ergemlidze,　Győri,　Methuku　[4]　that　ex3lin(n,C5)=Θ(n3/2).　In　this　paper,　we　strengthen　their　results　by　proving　that　[Formula　presented]　©　2022　Elsevier　B.V.