In this paper we investigate conditions for Lp-computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ⊂ ℝq, we show that an Lp function f ∈ Lp is LP-computable if and only if f is sequentially computable as a linear functional and the Lp-modulus function of f is effectively continuous at the origin of ℝq.