Agentic search requires large language models (LLMs) to perform multi-step search to solve complex information-seeking tasks, imposing unique challenges on their reasoning capabilities. However, what constitutes effective reasoning for agentic search and how it can be learned remains unclear. In this work, we first investigate the reasoning behaviors that enable success in agentic search. By comparing successful and failed trajectories via an LLM-based analysis pipeline, we identify four beneficial behaviors: Information Verification, Authority Evaluation, Adaptive Search, and Error Recovery. Building on this, we propose Behavior Priming, a training approach that equips agentic search models with these reasoning behaviors before reinforcement learning (RL). Specifically, it first performs supervised fine-tuning (SFT) on collected trajectories exhibiting the identified behaviors to cultivate these behaviors, and then applies standard RL to further improve task performance. Experiments on Qwen3-1.7B and Llama3.2-3B-Instruct show that Behavior Priming yields relative improvements over direct RL by 37.2\% on three web benchmarks and 6.2\% on seven multi-hop QA benchmarks, and outperforms the SFT-then-RL baseline using outcome-correct trajectories for fine-tuning. Crucially, we show that these reasoning behaviors matter more than outcome correctness in the priming stage prior to RL. Further analysis reveals that Behavior Priming enhances exploration (pass@8) and test-time scaling (search step number), providing a robust foundation for RL. Our code are avalible at https://github.com/cxcscmu/Behavior-Priming-for-Agentic-Search.