We present a high-resolution synthetic spectral library, INTRIGOSS, designed for studying FGK stars. The library is based on atmosphere models computed with specified individual element abundances via ATLAS12 code. Normalized SPectra (NSPs) and surface Flux SPectra (FSP) in the wavelength range 4830-5400 angstrom were computed with the SPECTRUM code. INTRIGOSS uses the solar composition of Grevesse et al. and four [alpha/Fe] abundance ratios, and consists of 15,232 spectra. The synthetic spectra are computed with astrophysical gf-values derived by comparing synthetic predictions with a solar spectrum of very high signal-to-noise ratio and the UVES-U580 spectra of five cool giants. The validity of the NSPs is assessed by using the UVES-U580 spectra of 2212 stars observed in the framework of the Gaia-ESO (European Southern Observatory) survey and characterized by homogeneous and accurate atmospheric parameter values and by detailed chemical compositions. The greater accuracy of NSPs with respect to spectra from the synthetic spectral libraries AMBRE, GES_Grid, PHOENIX, C14, and B17 is demonstrated by evaluating the consistency of the predictions of the different libraries for stars in the UVES-U580 sample. The validity of the FSPs is checked by comparing their prediction with both the observed spectral energy distribution (SED) and spectral indices. The comparison of FSPs with SEDs derived from the libraries ELODIE, INDO-U.S., and MILES indicates that the former reproduce the observed flux distributions within a few per cent and without any systematic trend. The good agreement between observational and synthetic Lick/SDSS (Sloan Digital Sky Survey) indices shows that the predicted blanketing of FSPs well reproduces the observed one, thus confirming the reliability of INTRIGOSS FSPs.