BE55 pilot reported a gear up landing.

Narrative

I departed with two passengers from ZZZ1 for ZZZ2 (home base) on an IFR flight plan. We planned an interim fuel stop at ZZZ. While approaching ZZZ; we had the field in sight; canceled IFR; and switched to the Unicom frequency. Upon reaching pattern altitude; I configured the airplane for landing; including lowering the landing gear. The landing gear sounded and felt like it lowered; however; there were no green lights indicating it was down and locked. We attempted to contact someone at the airport on the Unicom frequency so they could view the landing gear from the ground. We received no response. I had the airport phone number in my phone from an earlier call I had made prior to departure confirming they had fuel available. I called the number and the gentlemen who answered informed me the airport was unattended today. We decided to divert to ZZZ3; a nearby airport that we knew had a tower and emergency equipment. I contacted ZZZ3 Tower and informed them we wanted to request priority handling and divert there. We also requested a flyby so the tower could inspect the landing gear. Upon reaching ZZZ3; we conducted the flyby; where upon the tower controller informed us that the right main gear and nose gear appeared down; however; the left main gear was still retracted. We requested some airspace in the proximity of the airport to troubleshoot. We performed the emergency gear extension procedure; however; we did not receive any green lights. The landing gear motor breaker was tripped and would not reset. We were at a low enough altitude that I had cell phone coverage; so I conferred with our mechanic and also an engineer very familiar with Beechcraft landing gear systems. They both offered various suggestions; including putting some G forces on the airplane to see if it would force the gear down into a locked position. None were successful. The engineer informed us landing with two down and one retracted was a very dangerous situation; and the airplane would likely flip; and perhaps even cartwheel on landing. I prepared the passengers for this possibility; and we began our approach to land. While on final; my cell phone rang; it was the engineer calling. He told me he had an idea and to abort the landing; which we did. The engineer asked how much fuel we had; and I told him approximately two hours. He suggested we fly around for a while and give the landing gear motor time to cool; whereby it might run long enough to retract the gear. We followed his advice; and indeed it worked and the gear retracted. We asked for another tower flyby to confirm where upon the tower controller informed us that all three gear were now retracted; except for the left main gear door. Knowing that this was a much more favorable condition; we told the tower we wanted to land gear up. They cleared us to land; and we made an uneventful gear up landing.