IN the fiscally conservative and fiercely independent state of New Hampshire, a small but growing third political party is becoming more vocal.Well-organized and aggressively promoting its ideals, New Hampshire's Libertarian Party is considered at the forefront of the struggling Libertarian movement in the United States. "We're watching them very carefully to see if the lessons they learn up there can be applied elsewhere in the country," says Marc Montoni, administrative assistant for the National Libertarian Party in Washington, D.C. Founded in 1971, the Libertarian Party has a membership of about 250,000 registered voters. Its strongest followings are in New Hampshire, Maine, California, and Alaska. The party's platform combines both conservative and liberal beliefs. Libertarians favor limited taxes and greater individual freedoms. They believe that government should protect the country against force and preserve basic constitutional rights, but that most government-run services should be privatized. "Libertarians don't believe in utopia," says Bill Winter, chairman of the New Hampshire Libertarian Party. "We're just convinced free men and women can solve problems better than the bureaucrats in Washington." Libertarians say eloquent candidates and voters disenchanted with the Republican or Democratic parties have helped galvanize more support for the movement in New Hampshire during the last three years. They also say the state has a strong libertarian streak. "Given the general attitude of New Hampshire residents Live Free or Die they are essentially libertarian at heart," says Gene Burns, a radio talk-show host in Boston and a well-known Libertarian who speaks about the party's philosophy in New Hampshire and around the country. "They like a small government, not too costly, which doesn't impinge on personal freedom and doesn't engage in a lot of income redistribution." New Hampshire Libertarians point to several small successes in the state that have helped boost their party's standing: * In the 1990 gubernatorial election, the Libertarian candidate, Miriam Luce, received about 5 percent of the vote - enough to give the party legal status for the first time in the state's history. * Two months ago, State Rep. Calvin Warburton, a longtime Republican, switched to the Libertarian Party. According to party officials, he is the first state legislator in the country to change party affiliations to become a Libertarian and is now the only Libertarian state representative in the country. (Alaska had several Libertarian state representatives during the 1980s.) * A Libertarian selectman in Epsom, N.H., persuaded town officials to pass the nation's first education tax abatement plan, under which the town will refund property tax money to people who send their children to private schools. Libertarians say the passage of this plan shows the kind of solutions - and not just theories - their party can offer to problems. New Hampshire's Libertarian Party is concentrating its energies on helping two Libertarians campaign for two fill-in seats in the state House of Representatives. The party is also planning strategies for the 1992 elections. Mr. Winter says the party hopes to field about 100 candidates for the 400 state representative slots. In the past, only two or three candidates have run. In addition, the state will have its first Libertarian presidential primary. Despite the progress the party says its making in New Hampshire, some observers say it has little influence. "The House has 400 members. One or two people can't make much of a splash there unless they're in the power structure, and the power structure is Republican," says Robert Craig, a professor of political science at the University of New Hampshire. "They make no splash whatsoever electorally." Winter says: "We realize we're not going to change the country overnight ... , but I think people are willing to listen to new ideas from people like the Libertarians."