AbstractIt is proved that ifTis sufficiently large, then uniformly for all positive integersℓ⩽(logT)/(log2T), we havemaxT⩽t⩽2Tζ(ℓ)1+it⩾eγ·ℓℓ·(ℓ+1)−(ℓ+1)·log2T−log3T+O(1)ℓ+1,where γ is the Euler constant. We also establish lower bounds for maximum of|ζ(ℓ)(σ+it)|whenℓ∈Nandσ∈[1/2,1)are fixed.