I tried looking on Google, but there are some fairly contradictory results.
I thought I'd ask you guys so we could get an authoritative answer on the subject!
So in practice zeros is preferred in the US and even more so in the UK, though citations for zeroes include such prominent examples as Stephen Hawking's Brief History of Time. (Mr Hawking is British, but the book was first published by an American publishing house.) The oldest citation for zeroes in the BNC is from 1978. The Corpus of Historical American English has six citations that are even older, the oldest one being from 1914.
Note that dictionaries document the (current, at the time of going to press) usage of language, they aren't authoritative. 'Correct' is what is in common usage and largely understood to be correct, even if that contradicts a dictionary (in which case the dictionary is probably out-of-date).
So, as RegDwight has already answered, either zeros or zeroes is 'correct', but....
It's interesting to note that the Lexico's sole definition of zeroes is related to zero as a verb, e.g. "watch as he zeroes his sights on the target"; not as the plural of zero. It states that the plural of zero is zeros. This doesn't mean that using zeroes as a plural is wrong, as I've already said, it just shows that such usage is probably a more recent occurrence, gaining acceptance as the use of zero as a verb falls into decline.
In addition, the OED quotes zeroes as the only plural of zero. Both sites are run by Oxford Press, the former providing data from a collection of Oxford Press dictionaries. I think the difference between the two emphasizes my point that there is no 'correct', and that dictionaries merely document popular usage. I imagine there are more resources dedicated to the OED and that this is more likely to be up-to-date, but really that's just speculation.
I prefer "zeroes" because "zeros" resembles the Greek singular and seems to invoke the pronunciation ZEH-ross, and I'm not the only one. Oxford explains their pluralization rules including an appearance of zeros, here: Oxford Dictionaries: Plurals of Nouns.
At this point I am beginning to sympathize with Dan Quayle and his potatoe incident!
An examination of early instances of the plural of zero in English suggests that the more common spelling has long been zeros, which also seems to have appeared in English writing somewhat earlier than zeroes. Here is an Ngram chart for zeros (blue line) versus zeroes (red line) for the period 1760–1880:
I started at 1760 to avoid a number of results from French and Spanish sources—along with some false matches based on OCR errors that severely skew the line plots for earlier years. The vast majority of instances of zeros and zeroes from 1750 and later in Google Books results are from English texts.
The earliest instance in English of zeros that a Google Books search uncovers appears in a review of Some Familiar Letters Between Mr. Locke and Several of His Friends (1708) in The History of the Works of the Learned, Or, An Impartial Account of Books Lately Printed in All parts of Europe (January 1708):
In a Letter to Mr. Locke, (Aug. 24, 1695) in commendation of his Thoughts of Education, Mr. Molyneux tells him what wonderful Effects that Way has had upon his own Child ; who when he was turn'd five Years old could read perfectly well ; and on the Globes could have trac'd out and pointed at all the noted Parts, Countries and Cities of the World ; and by five and a half could perform many of the plainest Problems on the Globe, as the Longitude and Latitude, the Antipodes, the Time with them and other Countries, &c. and this by way of Play and Diversion, seldom call'd to it, never chid or beaten for it. About the same Age, he tells us, he could read any Number of Figures not exceeding fix Places, break it as you please by Cyphers or Zeros. By the Time he was six (continues he) he could manage a Compass, Ruler and Pencil very prettily, and perform many little Geometrical Tricks, and advanc'd to Writing an Arithmetick.
William Molyneux was born in Dublin and was third-generation Anglo-Irish, but he was also fluent in French (he translated Descartes' Meditations in 1680), so he may well have been familiar with zeros as the standard French spelling of the plural of zero. In any case, zeros is the spelling he uses in his 1695 letter to Locke, which appears in its entirety in volume 4 of The Works of John Locke, eighth edition (1777).
Additional instances in which native English speakers use the spelling zeros begin to appear in 1777. From William Roy, "Experiments and Observations Made in Britain, in Oder to Obtain a Rule for Measuring Heights with the Barometer" (read June 12 and 19, an November 6 and 13, 1777), in Philosophical Transactions of the Royal Society of London for the Year 1777 (1778):
...for the French toise bears to the English fathom, the proportion of 106575 to 100000; wherefore 6575/106575 = 61.69/2.1 = 29°.4 + 32° = 69°.4 denotes the relative positions of the two zeros, the intermediate equation 61.69/1000 being to be subtracted, when the toise is made use of. But it hath been shewn, that the mean expansion of air is really greater, for such temperatures at least as the barometer can be applied in, than what Mr. De Luc supposed it, in proportion of 24.5 to 210; whence it follows, that 61.69/2.45 = 25°.18 + 32° = 57°.18, will denote the relative positions of the two zeros; which, instead of almost 30°, are only distant from each other a little more than 25°.
And from David Stewart & Walter Minto, An Account of the Life, Writings, and Inventions of John Napier of Merchiston (1787):
In order that division may be performed by the Promptuary, it must first be converted into multiplication by means of tables dressed on purpose, or of tables of sines, tangents and secants, constructed on the hypothesis of the radius being equal to unity, followed by a certain number of Zeros. ... Find the product by the promptuary as above directed. This product, a number of the right hand figures according to the number of zeros in the square of the radius being marked off as decimals, is the quotient required.
I think it is even beyond doubt that Napier, in common with all other arithmeticians acquainted with the Arabic, or rather Indian figures, had observed that the product of any power of the number 10 by any other power of that number, was formed by joining or adding the zeros in the one to those in the other; and that the quotient of any power of that number by any other, was formed by taking away or defacing a number of zeros in the dividend equal to the number of zeros in the division; and all this without thinking that he was, at that time, making the fundamental remark of the logarithms.
Instances of the plural spelling begin to appear in the first decade of the 1800s. From a review of Asiatic Researches (1803) in The Critical Review: Or, Annals of Literature (August 1804):
If we suppose a cubic stone of nine cubits on each side, and a goddess, dressed in robes of the finest muslin, to pass near it once only every thousand years, with the zephyr gently blowing against the stone; the time in which the stone is worn down to the size of a grain of mustard forms the space styled antakalpe. Eighty antakalpes form a mahakalpe! The asanke, another period, comprehends a number of years expressed by unit and sixty-three or sixty-four zeroes. This is followed by some computations of peculiar curiosity, for which we must refer to the volume.
And from Maria Edgeworth, Patronage, volume 2 (1814):
"I have said so much more than I intended of both these brothers, that I have no room for more portraits, indeed, the other gentlemen are zeroes.
" * * * * * * * * * * * * * * * *
Google Books search results find published instances of zeros occur in Spanish texts as early as 1623 and in French texts as early as 1663. The English spelling zeros may have originated as an imported orthographical form. William Molyneux, the earliest English writer I could find who wrote a plural form of zero, was probably familiar with the French spelling zeros and may have based his spelling of the word on that.
Nevertheless, the home-grown English variant zeroes arose more than 200 years ago and has always had some adherents. Here is the Ngram chart for zeros (blue line) versus zeroes (red line) for the period 1760–2015:
Spot checks of the underlying search results indicate that the most of the matches for both spellings involve instances in which the writer is using zeros or zeroes as a noun. It seems pretty clear from this chart that zeros remains the dominant orthographical form in English but that zeroes continues to attract a substantial following, too.