Formal verification provides a rigorous and systematic approach to ensure the correctness and reliability of software systems. Yet, constructing specifications for the full proof relies on domain expertise and non-trivial manpower. In view of such needs, an automated approach for specification synthesis is desired. While existing automated approaches are limited in their versatility, i.e., they either focus only on synthesizing loop invariants for numerical programs, or are tailored for specific types of programs or invariants. Programs involving multiple complicated data types (e.g., arrays, pointers) and code structures (e.g., nested loops, function calls) are often beyond their capabilities. To help bridge this gap, we present AutoSpec, an automated approach to synthesize specifications for automated program verification. It overcomes the shortcomings of existing work in specification versatility, synthesizing satisfiable and adequate specifications for full proof. It is driven by static analysis and program verification, and is empowered by large language models (LLMs). AutoSpec addresses the practical challenges in three ways: (1) driving \name by static analysis and program verification, LLMs serve as generators to generate candidate specifications, (2) programs are decomposed to direct the attention of LLMs, and (3) candidate specifications are validated in each round to avoid error accumulation during the interaction with LLMs. In this way, AutoSpec can incrementally and iteratively generate satisfiable and adequate specifications. The evaluation shows its effectiveness and usefulness, as it outperforms existing works by successfully verifying 79% of programs through automatic specification synthesis, a significant improvement of 1.592x. It can also be successfully applied to verify the programs in a real-world X509-parser project.