Hey,
I recently had a problem with my shifter on the G25. When I tried going into 1st and 3rd gear it would only go into 5th, and when I tried 2nd and 4th it would only go in to 6th. I emailed Logitech and they have now sent me a whole new set because mine was under warrenty.
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Once my new one had arrived I took apart the old shifter and found a little black wire had come loose from the plug. You can see it in the pics below, it is meant to be soldered to the far right. That has now been soldered back in and I have 2 working G25s. I'm going to keep the new one in it's box and use my original till it breaks again.
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If yours is less than 2 years old try and get a replacement, otherwise take it apart and see what you can do. I'm very satisfied with Logitechs customer service and I would expect them to sort yours out.