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Aiming at exploring the nature of dark energy, we use thirty-sixobservational Hubble parameter data (OHD) in the redshift range $0 \leqslant z\leqslant 2.36$ to make a cosmological model-independent test of the two-point$Omh^2(z_{2};z_{1})$ diagnostic. In $\Lambda$CDM, we have $Omh^2 \equiv\Omega_{m}h^2$, where $\Omega_{m}$ is the matter density parameter at present.We bin all the OHD into four data points to mitigate the observationalcontaminations. By comparing with the value of $\Omega_{m}h^2$ which isconstrained tightly by the Planck observations, our results show that in allsix testing pairs of $Omh^2$ there are two testing pairs are consistent with$\Lambda$CDM at $1\sigma$ confidence level (CL), whereas for another two ofthem $\Lambda$CDM can only be accommodated at $2\sigma$ CL. Particularly, forremaining two pairs, $\Lambda$CDM is not compatible even at $2\sigma$ CL.Therefore it is reasonable that although deviations from $\Lambda$CDM exist forsome pairs, cautiously, we cannot rule out the validity of $\Lambda$CDM. Wefurther apply two methods to derive the value of Hubble constant $H_0$utilizing the two-point $Omh^2(z_{2};z_{1})$ diagnostic. We obtain $H_0 =71.23\pm1.54$ ${\mathrm{km \ s^{-1} \ Mpc^{-1}}}$ from inverse varianceweighted $Omh^2$ value (method (I)) and $H_0 = 69.37\pm1.59$ ${\mathrm{km \s^{-1} \ Mpc^{-1}}}$ that the $Omh^2$ value originates from Planck measurement(method (II)), both at $1\sigma$ CL. Finally, we explore how the error in OHDpropagate into $w(z)$ at certain redshift during the reconstruction of $w(z)$.We argue that the current precision on OHD is not sufficient small to ensurethe reconstruction of $w(z)$ in an acceptable error range, especially at thelow redshift

Utility of observational Hubble parameter data on dark energy evolution