This　paper　is　concerned　with　the　zero-relaxation　limits　for　periodic　smooth　solutions　of　the　non-isentropic　Euler-Maxwell　system　in　a　three-dimensional　torus　prescribing　the　well/ill-prepared　initial　data.　The　non-isentropic　Euler-Maxwell　system　can　be　reduced　to　a　quasi-linear　symmetric　hyperbolic　system　of　one　order.　By　observing　a　special　structure　of　the　non-isentropic　Euler-Maxwell　system,　we　are　able　to　decouple　the　system　and　develop　a　technique　to　achieve　the　a　priori　H-S　estimates,　which　guarantees　the　limit　for　the　non-isentropic　Euler-Maxwell　system　as　the　relaxation　time　tau　-＞　0.　We　realize　that　the　convergence　rate　of　the　temperature　is　the　same　as　the　other　unknowns　in　the　L-infinity(0,　T-1;　H-S),　but　the　convergence　rate　of　the　temperature　is　slower　than　the　velocity　in　L-2(0,　T-1;　H-S).　The　zero-relaxation　limit　presented　here　is　the　transport　equation　coupled　with　the　drift-diffusion　system.　However,　the　limit　of　the　isentropic　Euler-Maxwell　system　is　the　classical　drift-diffusion　system.　This　shows　the　essential　difference　between　the　isentropic　and　non-isentropic　Euler-Maxwell　systems.