In the present paper, we study a lifespan of solutions to the Cauchy problem for semilinear damped wave equations(DW) where , or , , is a small parameter, and is a given initial data. The main purpose of this paper is to prove that if the nonlinear term is and the nonlinear power is the Fujita critical exponent , then the upper estimate to the lifespan is estimated by for all and suitable data , without any restriction on the spatial dimension. Our proof is based on a test-function method utilized by Zhang [35]. We also prove a sharp lower estimate of the lifespan to (DW) in the critical case .