With their share of the vote at around 31% this time, according to the Irish Examiner/Prime Time poll, not even Fianna Fáil is suggesting it can take a second seat this time.
The closest Labour came to taking a seat in this constituency was in the 1994 election when Jim Fitzsimons (FF) needed a recount to take the last of the four seats from Labour's Michael Bell by just 10 votes.
The latest poll predicts another close call, but this time it will be between Labour's Peter Cassells and Fine Gael hopeful Máiread McGuinness.
She has 15% of first preference votes, while Mr Cassells has 14%. Both parties are well aware that the winner will be whoever can get ahead at this early stage.
Fianna Fáil took two of the four seats in the 1999 election despite winning almost the same number of votes as Fine Gael.
However, poor vote management saw FG's Avril Doyle top the poll just shy of a quota while her running mate, Alan Gillis, was eliminated. The two Fianna Fáil candidates Jim Fitzsimons and Liam Hyland were elected without reaching the quota.
Veteran Fine Gael strategist Phil Hogan hopes that better vote management from his party this time around will elect sitting MEP Ms Doyle and newcomer Ms McGuinness.
The trend shows Ms McGuinness is closing the gap on Ms Doyle. There are just 4.6 percentage points between them after the sitting MEP slipped back from the 22% she polled just over a week ago.
Labour is hoping it can stay ahead of Ms McGuinness, although the Irish Examiner/Prime Time poll suggests Mr Cassells is failing to find the additional first preference votes he needs if he is to come within striking distance of the 25% quota and edge into the third seat.
At the moment, the three main parties believe it's all to play for even though none of them sounds completely confident.
The losers are not just Fianna Fáil, but also the Greens. Their candidate, Mary White, is taking only 7.4% of the first preferences according to the poll, managing to stay just ahead of Sinn Féin's John Dwyer who is on 6.8%.
Ms White, from the north of the constituency, is not translating as well across the territory as did her predecessor, outgoing MEP Nuala Ahern. Ms Ahern, a founder of the Greens, had become a national name before running for the European Parliament and first won her seat in 1994.
Fianna Fáil takes 31% of the first preferences, according to the poll. This is down on their 33% core vote. Fine Gael, which is normally evenly matched with the governing party, is ahead with 34% of first preferences although this is still down on the 34.6% of the vote they got in 1999.
Three-seater constituencies usually do not suit Fianna Fáil, especially when their core vote is at the kind of levels it is in East. They are confident of Liam Aylward taking the first seat and, on current figures of 22.2%, he should top the poll. The party says it will try to boost TD Seamus Kirk's votes over the next two weeks, but with him standing at just 8.5%, the gap is very wide.
FF would like to keep Mr Kirk, who is chairman of the parliamentary party, in the running ahead of Máiread McGuinness.
Her TV farming profile on Ear to the Ground is expected to net her quite a few votes and it is thought these would return to Fianna Fáil if she was eliminated ahead of Mr Kirk. But Fianna Fáil would need a small miracle to achieve this now.
With the quota standing at 25%, Fine Gael estimates it needs between 36% to 42% of the vote to win an unprecedented second seat.
Avril Doyle, the only sitting MEP running for election in this constituency, believes her party needs to add about 7% to its core vote of 34% to be in with a real chance of taking the second seat.
The latest poll sees her share of the vote dropping back to 19%, down from a high point of 23%, while Ms McGuinness, whose rating had scarcely moved in earlier polls, has now jumped from 12% to 15% of first preference votes.
So far nobody is suggesting she will continue to take votes at the expense of her running mate, but Fine Gael will need to take a share from some of the other candidates if they are to be sure of a second seat.
Ms McGuinness, however, is being doggedly optimistic and says her aim all along has been to add a Fine Gael seat rather than to swap personnel in the seat they have. The key to achieving a double, however, is to outlast Labour and ensure Ms McGuinness benefits at least as much as the other candidates from transfers. Her TV profile should ensure she picks up votes from all over this vast constituency, but she needs to get them high enough on the ballot paper.
Labour has consistently had a core vote of 11% in this constituency and they are hoping that the profile of trade unionist Peter Cassells will add another 3 to 4% to this. However, even this will leave the former Irish Congress of Trade Union chief short of a realistic chance and so they are battling to add another 2% before the election.
Up to now, the polls have put him in third place ahead of Ms McGuinness, the Greens and Sinn Féin, though the FG candidate has taken a very narrow lead over him in the Irish Examiner poll. If she can maintain this, she can expect to benefit from transfers from the Greens.
Mr Cassells also suffers from being from the crowded end of the constituency but should Mr Kirk or Ms McGuinness go first, he could benefit from geographical factors.
The Greens are further behind in the polls than many expected, with Ms White taking 8% of first preferences. This could be partly due to Ms White's name being last on the ballot paper often a more significant factor in opinion polls than in an election when voters go into the polling booth ready to search the ballot paper for their preferred candidates.
However, she is now so far behind a percentage point or two will not make much difference.
Sinn Féin's John Dwyer is not expected to do as well as his fellow party member Pearse Doherty in North West and he has 6.8% according to the poll.
There will be considerable interest in how the independent Justin Barrett from Navan will do in the election. He has been controversial during previous referenda for his stance on abortion, his anti-immigration views and his links to fascist groups in Europe. He has polled slightly more than any of the other independents on the ticket but is still managing just 2%.