  The prompt emission of gamma-ray bursts remains mysterious since the
mechanism is difficult to understand even though there are much more
observations with the development of detection technology. But most of the
gamma-ray bursts spectra show the Band shape, which consists of the low energy
spectral index $\alpha$, the high energy spectral index $\beta$, the peak
energy $E_{p}$ and the normalization of the spectrum. We present a systematic
analysis of the spectral properties of 36 GRBs, which were detected by the
Gamma-ray Burst Monitor (GBM), simultaneously, were also observed by the Large
Area Telescope (LAT) and the LAT Low Energy (LLE) detector on the $Fermi$
satellite. We performed the detailed time-resolved spectral analysis for all of
the bursts in our sample. We found that the time-resolved spectrum at peak flux
can be well fitted by the empirical Band function for each burst in our sample.
Moreover, the evolution patterns of $\alpha$ and $E_{p}$ have been carried for
statistical analysis, and the parameter correlations have been obtained such as
$E_{p}-F$, $\alpha-F$, and $E_{p}- \alpha$, all of them are presented by
performing the detailed time-resolved spectral analysis. We also demonstrated
that the two strong positive correlations $\alpha-F$ and $E_{p}-\alpha$ for
some bursts originate from a non-physical selection effects through simulation.
