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We experimentally realize a quantum random number generator (QRNG) with randomness of each generated number certified by the Kochen-Specker theorem. Employing this novel certification scheme eliminates the necessity for input seed random numbers and lifts the non-locality requirement for the certified QRNG. Implementation on the base of superconducting qubits allows achieving a bit rate of 25 kBit/s, two orders of magnitude higher than previously reported certified QRNGs. These improvements make our result a major step towards realization a practical certified quantum QRNG with implications for digital security and modelling algorithms.