Massachusetts has launched an updated version of its Health Connector website as the nation's new health care law takes effect.

State officials said visits to the website were brisk Tuesday and, despite a few glitches, more than 400 people had begun the application process for insurance in the first few hours.

States across the country launched online marketplaces Tuesday as a result of the federal law.

Massachusetts already leads the nation with the highest rate of insured residents — in part because of the health care law that served as the blueprint for the 2010 federal law signed by President Obama.

That Massachusetts law created the nation's first online health care marketplace.

Gov. Deval Patrick said the state is now hoping to lead the nation in reducing health care costs.