  It is known that local zeta functions associated with real analytic functions
can be analytically continued as meromorphic functions to the whole complex
plane. In this paper, the case of specific (non-real analytic) smooth functions
is precisely investigated. Indeed, asymptotic limits of the respective local
zeta functions at some singularities in one direction are explicitly computed.
Surprisingly, it follows from these behaviors that these local zeta functions
have singularities different from poles.
