Medvedev reached the last eight at the French Open last week having never previously won a main-draw match in ParisWorld number two Daniil Medvedev made a losing start to his grass-court campaign, losing 7-6 (8-6) 6-3 to Germany's Jan-Lennard Struff in Halle.
The Russian, who made the French Open last eight last week, led 5-2 in the first, but could not convert set point.
The second set was beyond rescue once Struff had taken two early breaks for a 5-1 lead.
Medvedev is the highest ranked player world number 45 Struff has beaten in his career.
The 31-year-old German has been in decent form of his own on the clay, reaching his first Tour final in Munich before making the last 16 at Roland Garros.
Top seed Medvedev, a hard-court specialist who has made the final of the Australian and US Open, has not been beyond the third round in three previous entries to the Wimbledon main draw.
Third seed Alexander Zverev plays fellow German Dominik Koepfer in his own first-round match later on Tuesday.
Stefanos Tsitsipas was due to be the tournament's second seed but the Greek pulled out of the event after losing the French Open final to Novak Djokovic in five sets on Sunday.
Roger Federer, who has Halle as a mainstay of his Wimbledon preparations, opened with a win over Belarusian Ilya Ivashka on Monday.
The Swiss 20-time Grand Slam champion will be chasing a record-extending ninth Wimbledon title when the championships start on 28 June.
