My early F1 has no numbers on the back

I bought the watch new from an authorized local dealer around 1989, and it's an early stainless case 28mm F1, blue dial/bezel, stainless bracelet.

I'd like to have it serviced, but TAG USA's form online asks for the serial number and reference number from the back, but it has no such numbers.

Suggestions?

It needs a battery for sure, and has never been serviced. Hasn't worked in years, but worked fine for ages. Can I get the crystal replaced (it has some scratches and a small chunk missing along ghe outer edge, and I'd like a new dial and bezel (black, rather than blue). Will this be crazy expensive?

Thanks.

It appears to be the 370.508 as shown in the 1990 catalogue below. You can find the catalogues in the Calibre11 knowledge center (Calibre11.com).

As for the crystal that’s easily replaced by any general watch maker and the crystal they should be able to measure order or have in their stock.
I’d say the crystal will be about $15-20. Labour not too much. A battery can be thrown in for a few dollars. Maybe look for someone local to service it and spend $50-100 I’d guess for a quartz ETA movement as I assume is fitted in this.

I would not bother sending it to Tag personally, but it’s up to you.

Throw it on a NATO strap in matching colours and for not much money you’ll have it back on your wrist and looking smart.