In-memory computing (IMC) based on spin-logic devices is regarded as an advantageous way to optimize the Von Neumann bottleneck. However, performing complete Boolean logic with spintronic devices typically requires an initialization operation, which can reduce processing speed. In this work, we conceptualize and experimentally demonstrate a programmable and initialization-free spin-logic gate, leveraging spin-orbit torque (SOT) to effectuate magnetization switching, assisted by in-plane Oersted field generated by an integrated bias-field Au line. This spin-logic gate, fabricated as a Hall bar, allows complete Boolean logic operations without initialization. A current flowing through the bias-field line, which is electrically isolated from the device by a dielectric, generates an in-plane magnetic field that can invert the SOT-induced switching chirality, enabling on-the-fly complete Boolean logic operations. Additionally, the device demonstrated good reliability, repeatability, and reproducibility during logic operations. Our work demonstrates programmable and scalable spin-logic functions in a single device, offering a new approach for spin-logic operations in an IMC architecture.