Updated

The Tea Party-backed lawmaker who lost to longtime Mississippi Sen. Thad Cochran in the Republican primary formally challenged the election results on Monday, alleging widespread problems with the vote and calling on the state Republican Party to recognize him as the winner.

"The facts, they're on our side. The law is on our side," state Sen. Chris McDaniel said at a press conference announcing the challenge.

The challenge was weeks in the making, as McDaniel volunteers combed records across the state for irregularities following the June 24 runoff election. The case was filed before the Mississippi GOP state executive committee, McDaniel attorney Mitch Tyner said.

McDaniel will have to prove there were enough illegally cast votes to change the outcome or that the election was so sloppily handled its result is in doubt.

Mark Garriga, an attorney for the Cochran campaign, said in a statement allegations of voter fraud were "baseless."

"We look forward to holding the McDaniel campaign to the burden of proof that the law requires – and, we are dedicated to the defense of the votes of those Mississippians who voted on June 24 for Thad Cochran as their United States senator, an election which has been as thoroughly reviewed and examined as any in modern Mississippi history," the statement said.

Garriga told Fox40 the case is an opportunity to disprove McDaniel's allegations.

"They've waited six weeks to do this. I think that says a lot about the seriousness of their challenge," he said.

Mississippi voters don't register by party, but state law makes so-called crossover voting -- casting a ballot in one party's primary and another party's runoff in the same cycle -- a misdemeanor. Tyner said the campaign had found 3,500 instances of crossover votes, along with the 9,500 "irregular votes" and 2,275 "improperly cast" absentee ballots. It was not immediately clear what made the votes irregular, or how the absentee ballots may have been improperly cast.

Certified results show Cochran won by 7,667 votes, or 51 percent.

McDaniel claimed that Cochran's team got more than 40,000 Democrats to vote in the GOP primary, but "in so doing, mistakes were made." Some, he claimed, were "intentional."

"They asked us to put up or shut up, and here we are with the evidence," McDaniel said.

McDaniel has called the June 24 runoff a "sham" and excoriated Cochran for seeking votes from "liberal Democrats." Cochran says there's nothing wrong with seeking support from Democrats and independents -- it's something he's done for decades.

While Mississippi courts have ordered some new local elections, there has been no court-ordered do-over of a statewide election in at least the past six decades of records reviewed by The Associated Press. Tyner has said he had found no examples of a Mississippi court ever ordering a new statewide vote.

McDaniel finished 1,418 votes ahead of Cochran in the June 3 primary, when the runoff was triggered because neither candidate won 50 percent of the vote. In the runoff three weeks later, turnout jumped by 63,295 votes to 382,197.

Cochran drew significant support in the runoff in majority-black precincts in Hinds County, which President Obama won in 2012. Hinds County is home to the capital city of Jackson, and it the most populous county in the state. Cochran also fared well in majority-white precincts in the Jackson suburbs of Madison and Rankin counties, where GOP challenger Mitt Romney received the most votes in 2012.

The Republican nominee will face Democratic former U.S. Rep. Travis Childers and the Reform Party's Shawn O'Hara in the Nov. 4 general election.

The Associated Press contributed to this report.