The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the N -level Friedrichs model. The long-time asymptotic-forms of both S(t) and P(t) are obtained for an arbitrary initial-state extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t i4 -decay and a faster decay are realized depending on the initial states, where the latter is estimated as t i8 . The study on the unstable systems was initiated by the works of Gamow [1], who attempts to explain the exponential-decay law of radioactivity. This law is also observed in the atomic systems coupled with the electromagnetic (EM) field, and its theoretical description is now understood by the poles on the second Riemann sheet of the complex-energy plane [2]. However, in the middle of the last century, the deviation from the exponential-decay law was predicted by Khalfin [3] both for short times and for long times. Around the turn of the century, a short-time deviation was successfully observed [4]. On the other hand, the long-time deviation has still not been detected [5], even though expected for arbitrary unstable systems with the continuum of the lower-bounded energy spectrum. The main cause behind the matter could be ascribed to too small survival probability S(t) at such long times, that is the component of the initial state remaining in the state at a time t. Some of the unstable systems can be reduced into the Friedrichs model [6, 7], which allows us to investigate the decaying behavior concerning such processes as the spontaneous emission of photons from the atoms [8, 9], the photodetachment of electrons from the negative ions [9-12], and so forth. The traditional study of the model often resorts to the single lowest-level approximation (SLA) of the atoms or negative ions, and it could be actually verified as long as such a level is quite separate from the higher ones. However, the multilevel treatment of the model has a possibility of another advantage: the choice of coherently superposed initial-states extending over various levels. In fact, it can yield a variety of temporal behavior that is never found in the SLA [13-15]. Such multilevel effects on temporal behavior are still not well studied except for Refs. [13-16], and much less examined with respect to nonexponential decay at long times. however examined such a long-time behavior of the survival probability S(t), incorporating the initial-state dependence, based on the N-level Friedrichs model [17]. discussion, obtained were proved to be quite general and general class Restricting ourselves to the weak coupling cases, we clarified how the asymptotic form of S(t), that follows a power-decay law, depends on the initial states. In particular, we disclosed the existence of a special initial state that maximizes the asymptotic form of S(t) at long times, which could be desirable for an experimental verification of the power-decay law, and also the initial states that eliminate the first term of the asymptotic expansion of S(t). The latter implies that S(t) for such initial states can exhibit another power-decay law, which is faster than the usual one. These results mean that the long-time behavior is determined by not only the small-energy behavior of the form factors but also the initial unstable-states. Such relations between the initial states and the power decay law were already studied with respect to the asymptotic behavior of wave packets, both for the free-particle system [18] and for finite-range potential systems [19]. In the present study, we derive the long-time asymptoticform of the total probability P(t) in the basis of the N-level Friedrichs model, and show its dependence on the initial unstable-states explicitly. P(t) is the probability of the system to remain in the subspace spanned by the unstable states, and is useful as a candidate for experimentally measurable quantities other than S(t) (see, e.g., [13, 20]). We fist consider the initial state localized at the lowest level, to look over the SLA from the multilevel approach. Then, the difference from the result based on the SLA is found unlike that forS(t). It is also proved that there exist the initial state maximizing the asymptotic form of P(t) at long times, and also the initial states eliminating the first term of the asymptotic expansion of P(t). One can then understand that these initial states have the same roles for S(t). Moreover, we numerically con