The D.C. Lottery has the highest per capita sales of any lottery in the country, according to a new study, and its governing board issued a report yesterday showing that revenues for the last six months were 39 percent higher than for the same period last year.

The lottery board, which has weathered sharp criticism of its operations from Mayor Marion Barry and members of the City Council, will generate revenues totaling $113.8 million in the fiscal year that began Oct. 1, compared to $86.3 million last year, according to the report.

The city will net about $35.7 million, after covering the cost of the lottery's administration, advertising and prizes. That is about $10.4 million more than was netted the previous year, fiscal 1984, but still about $4.2 million less than was projected in the 1985 budget.

A study prepared by the Massachusetts State Lottery shows that lottery sales in the District between January and the first week of April averaged $3.45 a resident per week, a figure greater than that for Maryland and 15 other states surveyed.

"I think the figures look good," said Brant Coopersmith, chairman of the D.C. lottery board. "We've got good equipment, good contractors and we've got a population that knows the game and plays the game."

Douglass W. Gordon, the board's executive director, said the District's No. 1 ranking "shows that the District can more than hold its own, even though we are one of the newest lotteries."

However, a city official familiar with the lottery said yesterday that District's per capita sales figures are somewhat inflated because a large part of the lottery's business comes from persons who live outside the city, and are thus not included in the population.

The official, who asked not to be identified, also said that the improved sales figures are due largely to the continued popularity of the daily three-digit number game, while the lotto and instant games are doing poorly.

The D.C. Lottery got off to a troubled start three years ago, with an initial surge of interest in the games resulting in overly ambitious revenue forecasts.

The lottery board and its prime contractor were sharply criticized for mismanagement and excessive overhead costs that were cutting into the District's potential revenues. The awarding of contracts were delayed by litigation and political wrangling, and a telephone strike delayed the installation of automated game terminals.

Today, much of the political controversy has died down, although the board still has an occasional mistake. Last summer, the lottery board awarded a $100,000 prize instead of the correct $50,000 award. Then in March of this year the board was forced to give away three new cars instead of two because of a clerical error.

"There's room for improvement, and we'll focus on the instant game," said Coopersmith.

D.C. Budget Director Betsy Reveal said yesterday that "things certainly do look more positive," although the lottery's net revenues still lag behind the projections.

"They're better managed and organized," she said. "The lottery board and staff are working better together. They're doing a much better job of tracking sales . . . ."

In the first six months of the current fiscal year, the board recorded gross sales of $54.5 million and expenses of $37 million, for a net gain for the city of $16.1 million. During the same period a year ago, the city netted $10.1 million.

City officials were hoping to net about $3.9 million more during the first half of this fiscal year. The shortfall resulted from a decision by officials to increase expenditures for advertising early in the year and a miscalculation of the cost of prizes, according to an official.