In this study, the state formation control issue of general linear networked multi-agent systems (MASs) is considered. Combining self-triggered strategy with general event-triggered strategy, a novel fully distributed asynchronous mixed self- and event-triggered control strategy is proposed, which can make the MASs achieve the prescribed state formation structure. The control strategy proposed in this study does not depend on any global network information. Therefore, no matter how large scale of the network is, the control strategy is feasible. Meanwhile, the control strategy uses the sampled state information at triggering instants instead of the real-time state information, which efficiently eliminates continuous communication among agents. Different from the existing studies, the event-triggered detector starts to work if and only if the next self-triggering instant comes in the control strategy proposed in this article. Thus, the control strategy can save more communication resources between sensor and event-triggered detector within the same inter-event interval compared with the previous event-triggered strategies. In addition, the control strategy designs a exponential decay term in the triggering function to rule out the unexpected Zeno behavior and reduce the triggering number. Finally, the numerical simulation result of multi-robot formation is given, which demonstrates the feasibility and performance of the proposed control strategy.