Alternation rarameter δ dependences of the ground-state energy (GE) e (δ) and the gap Δ (δ) in the excitation spectrum of the spin Hamiltonian H = J ∑ l =1 N [1+(-1) l δ] [ S l x S l +1 x + S l y S l +1 y +σ S l z S l +1 z ] are calculated in two numerical methods, with interest in the exponents β of the GE and α of the gap which are defined as e (δ) - e (0)∝δ β and Δ (δ)∝δ α for small δ. First by means of the 2-level renormalization group calculation the exponents are obtained as β∼1.78 and α∼0.96 for the isotropic Heisenberg model (σ=1). Second by means of the finite chain extrapolation, β for σ=1 is obtained as 1.68 -0.36 +0.13 from the data of the extrapolated GE between δ=0.05 and 0.1.