National Bank of Washington, the District's third-biggest bank, yesterday reported that its first-quarter profits rose sharply to $1.33 million (97 cents a share), compared with $508,000 (37 cents) for the first three months of 1982.

Luther H. Hodges Jr., chairman of the bank, said, "The earnings turnaround which occurred in the second half of 1982 has continued in 1983" even though the spread between the bank's cost of deposits and the amount it earned on loans and other investments narrowed.

Earnings during the first half of 1982, as well as for all of 1981, were hurt by the large number of low-yielding municipal bonds and mortgages National Bank of Washington had in its portfolio at a time when interest rates and the cost of obtaining deposits were sky-high. The bank also was saddled with a large number of sub-par loans made by earlier managements.

Hodges was brought in by the comptroller of the currency to run the bank after a series of scandals became public. He spent his first two years as chief executive trying to work out the problem loans and sell off some of the low-rate assets in the bank's portfolio.

He was helped when interest rates began to nose-dive last June.

The bank had assets of $1.06 billion at the end of March, compared with $1.11 billion on March 31, 1982. Deposits were $852 million at the end of March, compared with $844 million the year before. Hodges said the bank's assets will hover around $1 billion until it obtains more capital. The bank cut the yearly dividend in half, to $1 a share, in order to add more of its earnings to its capital base.