Light transmission properties of one-dimensional sinc function photonic crystals containing a dispersive defect layer with negative refractive index are studied by transfer matrix method. The results show that one-dimensional sinc function photonic crystals containing a dispersive defect layer with negative refractive index have a broader forbidden-bands than one-dimensional cosine function photonic crystals containing the same defect layer. The forbidden bandwidth of this function photonic crystals is rapidly shrunk and narrowed with the increase of the refractive index nB(0)、nA(0) or half period thickness of the dielectric layer, and defect modes are disappeared. The forbidden bandwidth of this function photonic crystals is widened with the increase of angles of incidence, the defect modes are redshift with the forbidden-bands, and the structure of forbidden bands is also sensitive to changes on the position of a dispersive defect layer with negative refractive index. But the thickness variation of defect layers can not change the position and width of the forbidden band, and defect modes move toward the center of the band gap with the increase of the defect layer thickness. These results are important for the design of one-dimensional function photonic crystals.