In this work, we take all the color-antitriplet diquarks, such as the scalar, pseudoscalar, vector, axialvector and tensor diquarks, as the basic constituents to construct the local pseudoscalar four-quark currents without importing the explicit P-waves to implement the negative-parity, and investigate the mass spectroscopy of the pseudoscalar hidden-charm tetraquark states without strange, with strange and with hidden-strange in the framework of the QCD sum rules in a consistent and comprehensive way. We obtain the lowest mass 4.56±0.08GeV for the pseudoscalar tetraquark state with the symbolic quark constituents cc¯ud¯, which is much larger than the experimental value 4239±18−10+45MeV extracted by the LHCb collaboration.