He was hit several times, including once in the head, by bullets from a home-made 7.62 millimeter TT gun, which was left at the scene.
Laukroze was one of the Riga Regional Court's most senior judges.
State police chief Juris Reksna told reporters the killing - the first of a Latvian judge - was likely to have been the work of a hired assassin. But the possibility of it being a random attack was not being ruled out, he added.
"There is a chance it was picked up by surveillance cameras at a nearby parking lot," Reksna told a crowded news conference on Oct. 16. "A special task force headed by Riga criminal police chief Valdis Pumpurs has been put together to try to solve this crime as fast as possible."
Reksna said he knew of no death threats having been made against Laukroze.
What he described as a "pretty good" composite picture of the suspect was assembled after interviews with 10 witnesses, who also spoke of a getaway car waiting for the killer.
The man police are looking for is 30 to 35 years old and 180-185 centimeters tall. He was wearing a black jumper, light blue jeans and a dark baseball cap. No details about the car are currently available.
Justice Minister Ingrida Labucka told the Baltic News Service the killing was probably related to Laukroze's work, due to the fact that it took place in daylight and had been made to look like a bungled robbery.
At a news conference on Oct. 16 she announced that Laukroze's family would be paid 50,000 lats ($81,300) in compensation - the same amount paid to the family of Vjaceslavs Liscovs, a senior tax official who was killed in eastern Latvia earlier this year and whose killer has yet to be apprehended.Labucka also asked the public for any information which could lead to the arrest of the perpetrator and said any one coming forward with such information would be put under a witness protection program.
Laukroze often took on the most complicated and controversial cases and at the time of his death was hearing a criminal case against Daira Silva, former head of the Latvian Association of Fashion Models and the association's manager Rolands Priverts.
He was due to hear a fraud case against businessman Armands Stendzenieks.
Earlier this year he was chief judge in the case against members of the National Bolshevik movement who occupied the bell tower of Riga's St. Peter's Church with what turned out to be a fake grenade.
Top Latvian officials voiced indignation at the killing.
Arnis Lapins, spokesman for Prime Minster Andris Berzins, said it showed that, "The criminal underworld is at war with the state."
Prosecutor general Janis Maizitis urged the police to put all its resources into solving the crime as quickly as possible. He may have been hinting at the possibility that the killer could flee the country, possibly to Russia, as has been the case with a number of suspects in the past.
On hearing of Latvia's latest high profile killing, MP Oskars Grigs concluded the country was spiraling inextricably downward. "First a state revenue service official, now a judge - next it'll be ministers and lawmakers," Grigs told BNS.
Monty Akkeson, chairman of Latvia's Foreign Investors' Council, said such killings did Latvia's reputation no good. "It's very hard to predict how investors will react, but the more the bad news, the more investors evaluate the situation - it's never good when the judicial system is attacked."
A senior Interior Ministry official, Andris Staris, urged the media not to try to investigate Laukroze's killing but to hand any relevant information to the police.