There is a general consensus among scholars that the first first few books of the Septuagint were translated in the early- to mid-third century B.C.E in Alexandria.
In her concise little Introduction to the Septuagint, Jennifer Dines ponders,
Why were written Scriptures needed, or permitted, in Greek at a time when they were not, apparently, in Aramaic? Why could there not have been oral Greek paraphrases of the Hebrew Scriptures in Egypt which might eventually have resulted in written Greek ‘Targums’?
Her conclusion is intriguing:
Alexandria (assuming the whole enterprise to have started there) provided a literate, cosmopolitan culture, where ‘everyone who was anyone’ came to study, and where debating and writing were second nature. Alexandrian Judaism may have almost accidentally pioneered a new stage in the history of the Bible in response to the excitement of living in a educated milieu which expressed iteself in written words . . . This was a moment of creative genius from which the LXX emerged as something generically new: not quite like a legislative document, not quite like a metaphrase of Homer, not quite an exegetical rewriting, but exhibiting features of all these genres. The first translators made serious use of all of them as appropriate ways of rendering the holy books in use within their communities, and in doing so perhaps attracted the interest of the wider world as well.