Stick your current L1 firmware update file on the microsd card in the camera and power on with the new battery. You will see the leds in it flicker shortly after powering on if it needs a firmware update (whilst it's running). I'd do this to be sure if I were you.
It didn’t do it automatically for me. The app was telling me to do a battery update and one of my TB 48 batteries. The only way I could do it was downloading the firmware update on the SD card. My new battery I had just purchased is not asking for a update through the app. So I am assuming it doesn’t need it. I did save the firmware update on the SD card so if I need to do an update it’s already downloaded on the SD card