Demand from large corporate users drove up rents for Class A office space in the Route 128/Massachusetts Turnpike office market during the first quarter, according to a new report by commercial real estate brokerage JLL.

Average asking rents hit $33.83 per square foot in the quarter ending March 31, up 4.4 percent from the end of 2013, according to JLL’s Office Outlook report. Rents are now at the highest level since the fourth quarter of 2008, JLL vice president of research Lori Mabardi wrote.

The biggest leases signed during the quarter were 52,000 square feet by Intralinks at 404 Wyman St. in Waltham, 42,000 square feet by Raytheon at 880 Winter St. in Waltham and 30,000 square feet by Karyopharm Therapeutics at 75-95 Wells Ave. in Newton. During the last 12 months, the submarket has experienced 531,761 square feet of direct net absorption.

The vacancy rate in the 128/Mass Pike market is 10.1 percent, down 2.5 percent in the last 12 months.