In this paper, we present an efficient implementation for the analytical energy-dependent Green's function coupled-cluster with singles and doubles (GFCCSD) approach with our first practice being computing spectral functions of realistic molecular systems. Because of its algebraic structure, the presented method is highly scalable and is capable of computing spectral function for a given molecular system in any energy region. Several typical examples have been given to demonstrate its capability of computing spectral functions not only in the valence band but also in the core-level energy region. Satellite peaks have been observed in the inner valence band and core-level energy region where a many-body effect becomes significant and the single particle picture of ionization often breaks down. The accuracy test has been carried out by extensively comparing the computed spectral functions by our GFCCSD method with experimental photoelectron spectra as well as the theoretical ionization potentials obtained from other methods. It turns out the GFCCSD method is able to provide a qualitative or semiquantitative level of description of ionization processes in both the core and valence regimes. To significantly improve the GFCCSD results for the main ionic states, a larger basis set can usually be employed, whereas the improvement of the GFCCSD results for the satellite states needs higher-order many-body terms to be included in the GFCC implementation.