In a previous work, the first named author described the set 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 α in 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 ω for all k in {0, .., n − 1} and equal to α for k = n. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to ω.