Direct Numerical Simulations (DNS) are carried out to investigate the primary and secondary instability for boundary layers on a straight (right) cone with a 7◦ opening half-angle at Mach 10 and zero angle of attack. The cone geometry of the experiments in the Arnold Engineering Development Complex (AEDC) Hypervelocity Wind Tunnel No. 9 (Tunnel 9) was used for the numerical investigations. Three different unit Reynolds numbers were considered for the primary and secondary instability calculations. As expected, the linear amplification rates and the corresponding N-factors decrease with decreasing unit Reynolds numbers. For all investigated cases the axisymmetric second mode disturbances were the dominant primary linear instability. An investigation of the primary wave saturation amplitudes revealed that the maximum second mode amplitudes depended only weakly on the unit Reynolds number. Secondary instability investigations showed that the wave angle of the secondary disturbance wave resulting in the strongest resonance (largest N-factor after resonance onset) remains largely unaffected by the unit Reynolds number. In addition, the same azimuthal wavenumber range experiences strong secondary instability for the investigated unit Reynolds numbers. Emphasis of the present paper is on the primary and secondary wave regime. Results from high-fidelity DNS of the entire transition process, from the linear stages all the way to complete breakdown, will be presented and discussed in a future paper.