We use integrable and conformal quantum field theories to compute the single-particle spectral function of armchair-type carbon nanotubes away from half-filling. The gapless charge sector is described by the Luttinger model and is solved by using conformal field theory. The other sectors are known to possess SO(6) symmetry at low temperature and are described by the SO(6) Gross-Neveu model, The correlation function of this massive sector is computed using exact form factors, Due to intermediate particle states, the spectral function is exactly determined by the given form factors between certain threshold energies. We derive. an exact expression explicitly for certain values of the Luttinger parameters at low frequency to illustrate the power of this approach.