  We investigated the time lags and the evolution of the cross spectra of Z
source GX~5-1, observed by the Rossi X-ray Timing Explorer (RXTE), when it is
in the horizontal branch oscillations. We showed that the time lags of 3
horizontal branch oscillations are related to the position on the hardness
intensity diagram. All of the three QPOs were shown to have hard time lags.
However on the cross spectra, one is in a `dip', one in a `bump', the other has
no so obvious characteristic. The time lags of two of the QPOs decrease with
QPO's frequency, while the other has a trend increasing with its frequency.
Moreover, in the normal branch, we found no significant time lags in the
present observational data.
