Given a partition λ = (λ_{1}, λ_{2}, ..., λ_{n}) denote with s_{λ} the associated Schur function.
There exists a nice product formula for the principal specializations:

s_{λ}(1, q, q^{2}, ..., q^{n-1}) = Π_{i<j} (q^{λi+n-i} - q^{λj+n-j}) / (q^{j-1} - q^{i-1}).

Is a similar evaluation known for specializations of the type s_{λ}(1, 2, ..., n)?