Given a partition λ = (λ₁, λ₂, ..., λ_n) denote with s_λ the associated Schur function. There exists a nice product formula for the principal specializations:

s_λ(1, q, q², ..., q^n-1) = Π_i<j (q^λ_i+n-i - q^λ_j+n-j) / (q^j-1 - q^i-1).

Is a similar evaluation known for specializations of the type s_λ(1, 2, ..., n)?