\begin{table}%t1
\caption{\label{tab:LF}\cite{1976ApJ...203..297S} function parameters
  for the external LFs.}  
%\centerline
 {\begin{tabular}{lcc}
\hline \hline\noalign{\smallskip}
Reference & $M_*$ & $\alpha_{\rm LF}$$^a$\\
\multicolumn{3}{c}{$z\sim3$}\\
\hline
\cite{1999ApJ...519....1S} & $-21.00$ & $-1.60$ \\
\cite{2006ApJ...642..653S} & $-20.90$ & $-1.43$ \\
\cite{2007ApJ...670..928B}$^b$ & $-20.98$ & $-1.73$ \\
\hline
\multicolumn{3}{c}{$z\sim4$}\\
%\hline
\cite{1999ApJ...519....1S} & $-21.20$ & $-1.60$\\
\cite{2006ApJ...642..653S} & $-21.00$ & $-1.26$ \\
\cite{2007ApJ...670..928B} & $-20.98$ & $-1.73$ \\
\hline
\multicolumn{3}{c}{$z\sim5$}\\
%\hline
\cite{1999ApJ...519....1S}$^b$ & $-21.20$ & $-1.60$$^c$ \\
\cite{2006ApJ...642..653S}$^b$ & $-21.00$ & $-1.26$ \\
\cite{2007ApJ...670..928B} & $-20.64$ & $-1.66$ \\
\hline
\end{tabular}}
\medskip
$^a$ This $\alpha_{\rm LF}$ is the faint-end-slope of the Schechter LF. The $\alpha(m)$ introduced in Eq.~(\ref{eq:flux_power_law}) approaches this value for faint magnitudes; $^b$  extrapolated from $z \sim 4$; $^c$  fixed. 
\end{table}