I don't know if I understood the doubt you had , anyway I will try to answer.
The stability in H(s) depends only of poles, if all poles is in the left side of the s plane the system is stable, if you have any pole in the right side the system is unstable. Zeros don't matters for stability, unless you have to take the inverse of the system, if so, in order to have a system causal and stable, it is necessary that all zeros and poles are in the left side of the s plane, such systems are called minimum phase.