PA-34 instructor pilot reported a main landing gear indication malfunction while on initial approach during a training flight. The instructor landed safely and the fault was determined to be the gear indicating system.

Narrative

On Day 0 during a flight from approximately XA00-XC00; as an instructor with a student; I experienced an abnormal gear indication. Upon placing the gear switch in the down position; I received an indication of two green lights (nose; right main) and one not down indication (left main). This occurred on the left downwind of runway XX at ZZZ [airport]. As PIC; my initial reaction was to check circuit breakers regarding a possible electrical issue; followed by checking normal electrical indications and instruments such as the ammeters for both alternators. When it was confirmed that there was nothing abnormal electrically; I designated my qualified student; who holds a commercial multi engine pilot certificate as the pilot flying while I took on the jobs of communications to relay intentions on CTAF; as well as troubleshooting the abnormal indication. The first troubleshooting method was to allow the gear (already down with abnormal indication) to be cycled back up. Once confirmed up with normal up indications; I placed the gear switch to the down position once more. The same abnormal down indication was present once again. I followed this up by exiting the traffic pattern on said downwind to the west and communicating the issue and my intentions with other air traffic on the CTAF. While buying myself time and space away from other aircraft; I confirmed navigation lights were off; which although necessary to confirm; could not explain why two gear indicator lights were full bright. This was then followed by replacing the two green lights into the left main gear circuit to see if the left main lightbulb had possibly died. This did not warrant either operating green light to turn green while in place of the left main lightbulb. After these troubleshooting techniques were attempted; the initial thought of mine while being unable to confirm; was most likely the circuit in which the left main lightbulb was situated; was blocked or inoperative; or there was a problem elsewhere within the electrical system that may have been preventing a green light from appearing in the cockpit. However I still had to remain cautious that the left main gear was either partially or not fully extended completely and that there was a possibility of the left main down limit switch not being locked in place. However; because of the absence of a gear warning unsafe light being illuminated; as well as the absence of a gear warning horn below throttles being placed below 14' MP; I was still under the impression of an electrical malfunction simply preventing a left main gear down indication. I then entered the downwind while communicating with the CTAF; and came in for a low approach on ZZZ [airport] runway XX; coordinating with an aircraft on the ground to perform a visual inspection of the extended landing gear. The aircraft promptly relayed that it appeared all gear was down and locked from their vantage point. A missed approach was conducted and I proceeded on a left crosswind to the north of ZZZ.Nevertheless; I decided that at this point in the flight; away from ZZZ traffic; and with gear remaining down; that the best course of action would be to alert the controlling agency; ZZZ Approach; of the abnormal; situation and continue an attempt at troubleshooting to confirm the down and locked status of the left main gear. Upon initial contact with ZZZ [approach]; and the roles within the cockpit still the same; I identified my aircraft and position as well as a thorough description of the abnormal indication at hand. I then decided that a low approach at ZZZ1 [airport] was plausible and a good idea with about 2.5 hours of fuel (60 Gallons) indicated. The controller vectored me on to the final approach course for ZZZ1 [airport] Runway XYL. I flew over the runway at approximately 50 feet with the gear extended; and was told by both the tower and a grounds crew vehicle that the gear did appear down. However; an aircraft short of the runway accounted that theleft main gear 'may have been slightly canted or tilted inwards' indicating it was not locked down. After executing a missed approach; it was my PIC decision to bring the aircraft directly back to ZZZ [airport] where maintenance service was available and the winds favored a large and wide runway (Runway XZ). I also [requested priority handling] as a precaution. At this point; uncertain of the status of the left main gear and enroute to ZZZ; I considered the possibility that the left main gear would remain unlocked and the potential dangers of landing with a main gear not fully down and it's consequences on directional control and possible fire hazard. I also still considered the possibility and less severe malfunction of a left main gear indicator; as an aircraft on the taxiway next to a runway cannot fully confirm a gear as down and locked or not. While enroute to ZZZ still; and remaining on ZZZ Approach frequency with a new controller; I decided that by bringing the gear up; turning the master switch off; and slowing below 100 MPH; then bringing the gear switch back to down; that I would pull the emergency gear extension. After doing this and the gear being heard free falling down; I still did not receive a green light for the left main gear. I allowed the gear switch to remain down as well as the emergency gear extension to remain pulled out for the duration of the flight.I notified ZZZ approach of the situation remaining the same and repeated my intentions of landing at ZZZ and was notified that fire and rescue was on the way to the airport. After switching back to the ZZZ CTAF; I communicated with a helicopter in the area; and the helicopter positioned itself on my right side and below; and did a final check; accounting that the gear did appear to be all down and locked. With this final account; I determined that with the helicopter being the closest aircraft with the best view; the absence of a gear warning unsafe light and the absence of the gear warning horn being present with any of the situations that it would present itself; that the most likely issue was an electrical indication regarding the left main gear and that it was in fact locked down. Now; north of ZZZ; I took controls of the aircraft and proceeded for a straight in approach to runway XZ; while in communications with an ops vehicle holding short of runway XZ; and confirmation emergency vehicles were standing by. I briefed my student on exit procedures in the event of a gear collapse upon landing. I had a light wind; headwind approach and landed with a sideslip technique to touch down right main first. This was followed by a safe touchdown of the left main and nose gear. I was prepared to shutoff fuel and air to both engines in case of a collapse with the left main gear never indicating locked down in the cockpit. I taxied straight down runway XZ to allow myself a minimum amount of turns back to a safe ramp on the airport and taxied without issue to a tie down.In summary; I used my knowledge of the Seneca I POH and systems; as well as resources within and outside the aircraft to allow myself to keep my students safety the number one priority and while attempting the most practical and thorough approach to a troubleshooting of the issue and a safe landing and outcome. I am thankful for the ZZZ TRACON help as well as other aircraft in the nearby airspace for their accommodations and assistance.