  In a previous work, the first named author described the set $\cal P$ of all
values of the Szlenk indices of separable Banach spaces. We complete this
result by showing that for any integer $n$ and any ordinal $\alpha$ in $\cal
P$, there exists a separable Banach space $X$ such that the Szlenk of the dual
of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$
in $\{0,..,n-1\}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to
show that the Lindenstrauss space and its dual both have a Szlenk index equal
to $\omega$. We also show that any element of $\cal P$ can be realized as a
Szlenk index of a reflexive Banach space with an unconditional basis.
