We propose a novel graphene-on-silicon organic hybrid slot microring resonator (GSHMIR) for tunable all-optical logic gates, which could offer broadband ultra-flattened dispersion with multiple zero-dispersion wavelengths (ZDWs) and high nonlinearity coefficient of 1.67 x 10(4) W(-1)m(-1). Further investigations on temporal evolution dynamics for all-optical logic function suggest that the third-order dispersion will induce the temporal drift and tail oscillation of output pulses. The tunable all-optical AND logic function over the wavelength range from 1536 nm to 1569 nm could be obtained for 40 Gb/s signals with the conversion efficiency as high as -6.4 dB. These results are essential for all-optical wavelength division multiplexer (WDM) technology of future integrated optoelectronic systems, as well as providing a novel architecture for on-chip optical communications.