For d≥ 2, p≥ 1 and ε> 0, let Np(d,ε) be the smallest integer N such that every d-dimensional subspace of Lp[0,1] admits a linear embedding into ℓpN with distortion at most 1+ε. For fixed d and p, the bound Np(d,ε) = \widetildeOd,p(ε-2(d-1)/(d+2p)) is established. For p∉ 2ℤ, this is optimal up to logarithmic factors; for positive even integers p, isometric embeddings of dimension independent of ε are known. The stated upper bound was previously known only for integer p. The case of non-integral p is handled by approximating |t|p by a polynomial with a remainder of small total variation; the error contributed by this remainder is controlled by equatorial band discrepancy.