Resetting alignment in the align environment

For full control of the alignment you can use the alignat environment.

\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{mathtools}   % loads »amsmath«

\begin{document}
  \begin{alignat}{4}
        &          & \gamma_1  &= 8/15\qquad & \gamma_2 &= 5/12         & \gamma_3 &= 3/4 \\
        &          & \zeta_1   &= 0          & \zeta_2  &= -17/60\qquad & \zeta_3  &= -5/12 \\
        &          & \beta_1   &= 4/15       & \beta_2  &= 1/15         & \beta_3  &= 1/6 \\
    a_0 &= 0\qquad & a_1       &= 8/15       & a_2      &= 2/3          & a_3      &= 1
  \end{alignat}
\end{document}

---

You can play around with columns to fool the alignment mechanism as:

\begin{align}
   \gamma_1 &= 8/15 &&& \gamma_2&= 5/12         &&& \gamma_3 &= 3/4 &\\
   \zeta_1  &= 0    &&& \zeta_2&= -17/60        &&& \zeta_3  &= -5/12& \\
   \beta_1  &= 4/15 &&& \beta_2&= 1/15          &&& \beta_3  &= 1/6&\\
     a_0 ={}&0      &&& a_1 = 8/15 && a_2 = 2/3 &&&\qquad a_3 = 1
\end{align}

but with all due respect, what you want to do, does not read well. I would have done something like the following

\begin{align}
   \gamma_1 &= 8/15 & \gamma_2&= 5/12  & \gamma_3 &= 3/4 \\
   \zeta_1  &= 0    & \zeta_2&= -17/60 & \zeta_3  &= -5/12 \\
   \beta_1  &= 4/15 & \beta_2&= 1/15   & \beta_3  &= 1/6\\
        a_0 &= 0    & a_1    &= 8/15   & a_2      &= 2/3 \\ \notag
            &       & a_3    &= 1      &          &
\end{align}

You can deliberately put some vertical space between the first three and the last to highlight the grouping effect which is, in my opinion, more important in order to convey the message rather than to be able to command a LaTeX array.

Here is one viable alternative to your alignment issue:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}
\begin{align}
   \smash{\begin{array}[t]{*{2}{r@{\;}l@{\qquad}}r@{\;}l}
     \gamma_1 &= 8/15 & \gamma_2 &= 5/12   & \gamma_3 &= 3/4 \\[\jot]
     \zeta_1  &= 0    & \zeta_2  &= -17/60 & \zeta_3  &= -5/12 \\[\jot]
     \beta_1  &= 4/15 & \beta_2  &= 1/15   & \beta_3  &= 1/6 
   \end{array}} \label{gammas} \\ \label{zetas} \\ \label{betas} \\
  \begin{array}[t]{*{3}{r@{\;}l@{\qquad}}r@{\;}l}
    a_0 &= 0 & a_1 &= 8/15 & a_2 &= 2/3 & a_3 &= 1
  \end{array} \label{as}
\end{align}
$\gamma$ is~\eqref{gammas}, $\zeta$ is~\eqref{zetas}, $\beta$ is~\eqref{betas} and $a$ is~\eqref{as}.
\end{document}​

The equation elements are placed inside an array. Proper vertical alignment is assured using \\[\jot], while horizontal adjustment is obtained via \qquad for the column spacing. Modify this to suit your need (to say \quad, or \hspace{<len>} where <len> is any known TeX length unit).