@Preben Alsholm 

Try this:

lambda = 25.5904477612, p = 0.0484865622, n= 27. For Digitst = 250 it gives all the 27 roots but it does not work for Digits=500 for instance. For every set of parameters there seems to be a given range of Digitst for which it works!

like this :

q := 1 - p;

f := z ^ n - exp(lambda * (z - 1)) * (q + p * z) ^ n;

zeros := [RootFinding:-Analytic(f, z, re = -1 .. 1, im = -1 .. 1)];

where n is a positive integer, lambda is a positive real, and p is real between 0 and 1. For lambda/nq < 1,  f should have n roots on or within the unit circle.