Organic semiconductors have promising and broad applications in optoelectronics. Understanding their electronic excited states is important to help us control their spectroscopic properties and performance of devices. There have been a large amount of experimental investigations on spectroscopies of organic semiconductors, but theoretical calculation from first principles on this respect is still limited. Here, we use density functional theory (DFT) and many-body Green's function theory, which includes the GW method and Bethe-Salpeter equation, to study the electronic excited-state properties and spectroscopies of one prototypical organic semiconductor, sexithiophene. The exciton energies of sexithiophene in both the gas and bulk crystalline phases are very sensitive to the exchange-correlation functionals used in DFT for ground-state structure relaxation. We investigated the influence of dynamical screening in the electron-hole interaction on exciton energies, which is found to be very pronounced for triplet excitons and has to be taken into account in first principles calculations. In the sexithiophene single crystal, the energy of the lowest triplet exciton is close to half the energy of the lowest singlet one. While lower-energy singlet and triplet excitons are intramolecular Frenkel excitons, higher-energy excitons are of intermolecular charge-transfer type. The calculated optical absorption spectra and Davydov splitting are in good agreement with experiments. [ABSTRACT FROM AUTHOR]