We investigate axisymmetric black hole (BH) formation and its gravitational wave (GW) and neutrino signals with self-consistent core-collapse supernova simulations of a non-rotating 40 M_ progenitor star using the isotropic diffusion source approximation for the neutrino transport and a modified gravitational potential for general relativistic effects. We consider four different neutron star (NS) equations of state (EoS): LS220, SFHo, BHBΛϕ and DD2, and study the impact of the EoS on BH

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... n dynamics and GW emission. We find that the BH formation time is sensitive to the EoS from 460 to >1300 ms and is delayed in multiple dimensions for ∼ 100-250 ms due to the finite entropy effects. Depending on the EoS, our simulations show the possibility that shock revival can occur along with the collapse of the proto-neutron star (PNS) to a BH. The gravitational waveforms contain four major features that are similar to previous studies but show extreme values: (1) a low frequency signal (∼ 300-500 Hz) from core-bounce and prompt convection, (2) a strong signal from the PNS g-mode oscillation among other features, (3) a high frequency signal from the PNS inner-core convection, and (4) signals from the standing accretion shock instability and convection. The peak frequency at the onset of BH formation reaches to ∼ 2.3 kHz. The characteristic amplitude of a 10 kpc object at peak frequency is detectable but close to the noise threshold of the Advanced LIGO and KAGRA, suggesting that the next generation gravitational wave detector will need to improve the sensitivity at the kHz domain to better observe stellar-mass BH formation from core-collapse supernovae or failed supernovae.