When you dial 623-6000, the D.C. government's main number what you get, sooner or later, is one of nine operators at an ancient spaghetti-style switchboard in a fifth floor room at the Municipal Center building, 300 Indiana Ave. NW.

The switchboard, which resembles a gigantic upright piano and bears the "Western Electric" emblem on a small brass plate, has been there since 1941, or 18 years before chief operator Alma Denton went to work at it.

If you want help tracking down a particular D.C. office or employee, the operator may consult any of the following tools of reference:

Her memory.

Her index card file of frequently called numbers.

The collective memories of the other operators in earshot.

Random notices sometimes paper-clipped or scotch-taped to the face of the switchboard - for instance, "LD calls are not to be accepted from . . .from any extension or other District telephone. He is . . . not authorized to make the calls."

If these aids fail, the operator will transfer a caller to D.C. government information head-quarters, about three yards away on the opposite wall. There may be up to four information operators on duty at any given time, each stationed by a hiking metal Rolodex file that looks like an upturned garbage can, containing 28,000 agency and employee phone numbers.

By contrast, each of the operators who answer 655-4000, the federal government's principal Washington-area number, sits at her own push-button "Centrex" console, where she can instantly flash about 90,000 federal phone numbers - indexed by employee, by office, and by function - onto her own personal TV screen.

The District's system is full of antique charm and potential for error. To plug an incoming party into the wrong hole is simplicity itself. Likewise it is equally as easy to sever two parties in mid conversation, or to leave a caller on indefinite hold.

To the ear at the other end, it sometimes may sound as if the operator is grinding away at her microphone with a steel brush. In fact, those scratches and scrapes and clicks are mere byproducts of the process by which a D.C. operator must seek an open line. She has to tap the inside edge of each hole with a metal plug, then listen to find out if a line is busy.

Operators also must contend with ill-tempered and disorganized people both outside and inside the halls of government. Many callers state a gripe and leave it to the operators to determine which of the District's 40,000 employees is best suited to tackle the problem.

D.C. operators undergo no formal instruction in the machinery or local government. It's "pure exposure" and a "Baptism of fire," according to the system's boss, T. J. (Dutch) Behrens, who was a telephone company salesman before coming to work for the District in 1971.

An automated information system for the District is nowhere in sight, says Behrens.' At the moment, the federal government's Washington-area telephone network, according to the people who run it, has the only such system outside the telephone company itself.

Behrens has begun the gradual process of phasing out the switchboard in favor of push-button Centrex consoles. He says the change will result in faster answering and connection of calls, and eventually in a reduced work force. To date, only the public schools, the D.C. Bail Agency, and a few units of the Department of Human Resources have been tied into the District's new Centrex network, with its new phone number, 727-1000.

Federal government operators, unlike those employed by the District, are put through a week of indoctrination before they work with the public. They are taught what the major executive agencies do and how each is organized: the ins and outs of the Centrex consoles and automated directory systems: and how to handle such touchy problems as bomb threats (they call the "control center" for the building affected).

It is mostly the equipment itself, which explains why federal operators often are able to provide astonishingly quick service. There are no cables, no holes, and no files. Once two parties have been connected, the operator does not have to disconnect them, manually at conversation's end.

Don Hardesty, director of the GSA's telecommunications division, feels his space-age phone technology has fired the enthusiasm of federal government operators. "It's a psychological thing. They're almost competing amongst each other to see who can do it the fasiest."

All of the D.C. and Federal government operators are women. Most are young. Despite Hardesty's bright picture of work-force morale, many of his present and former operators - and many of those employed by the D.C. government - seem to view the job as temporary. They say they intend to resign at the first opportunity in favor of a new occupation, school, marriage, or some blend of these.

Dorothy White, a federal operator who plans to leave in October, complains of under-staffing. Working in the phone room, she says, means non-stop handling of calls for the maximum of 40 operators on duty at peak hours. Operators are pressured to stick to their consoles for at least 400 minutes close to seven hours) per tour of duty, says White, and "necessaries" (trips to the bathroom) are limited to two of no longer than 3 minutes each.

Sandra Guthrie, now with C&-P Telephone, says she is much happier than during her year with the federal government. "They don't watch you the way they do in the government," Guthrie claims. And she does not mind taking home $80 more every two weeks, either, she says.

Out of a group of about 20 federal operators hired in July, 1976, according to White and Guthrie, all but five have quit distressed by what they regard as low pay and high pressure.

Hardesty denies that his operators are overworked. "I can assure you that conditions at the phone company are much more stringent he says. But he acknowledges that he has had trouble both finding and keeping operators, a problem he attributes to the pay.

Federal operators, like the District's can work their way up to a top salary of $8,500, excpet for the few who succeed in breaking through to the rank of supervisor.

A local official of the Communications Workers of America - the union that represents phone company operators - describes the job as "very, very confining." Operators "are under a lot of pressure," she says.

But she add, "I was an operator for 30 years and it didn't bother me any . . . there are always people who are notas strong."