There is at least one episode where it is shown; does anyone know what it is?

Mark

I got no clue, beside that all numbers start with a 0 s far as I know.

Season 4 seemed to show a few visible phone numbers. I don't remember seeing Cooter's number, but here are three from other threads.

In 'Coltrane vs Duke', the old Hazzard Hotel has a "For Lease" sign with the number 555-2359 (it's advertised as a light manufacturing property).

In 'The Great Bank Robbery', the "For Lease" number for J D Hogg is 555-4309.

In 'The Law and Jesse Duke', J D Hogg's "Hazzard Realty" number is 555-2133.

OK, here it is from 'Coltrane Vs. Duke' in season 4 (more info after the picture):

cooter's phone by mommydawn2000, on Flickr

Turns out that particular number has an interesting history. From Wikipedia:

"In the United States, 311 was sometimes used as a fictitious area code in Bell System advertisements depicting telephones; often the phone in the advertisement would bear the specific number "Area Code 311 555-2368." This fictitious phone number was used in the 1979 horror film When a Stranger Calls; in the opening of the television series The Rockford Files; in the 1984 film Ghostbusters; and on two episodes of the second season of the TV series Mission: Impossible: episode 22 titled "The Killing" (on the killer's phone) and, fourth season, episode 12, titled "Time Bomb" (on a phone at a nuclear plant in a fictitious country).

311 555-2368 was also used in numerous episodes of The Bionic Woman as the private phone number of Jaime Sommers in her coach house. At the end of a 1963 episode of Route 66, titled "Kiss the Monster, Make Him Sleep," Linc calls his estranged father at a number having area code 311. The number 311-555-9845 was used for a radio station hotline in episode 1, season 1, of the TV show A.L.F. This area code is also used as the area code for Sunnyvale, California (which is actually 408) in the 1983 movie WarGames during the "war dialing" sequence where the main character is searching for a video game company's modem pool.

"

Hazzard 741.    That's the number Luke asks for when  phoning  Cooters,in "Play it Again,Luke ",  S6,episode 15

In The Treasure of Soggy Marsh the number to the Duke farm is " Hazzard 587"

That's cool! I never knew this stuff!!