I couldn't find any clear rules/guidelines regarding this, so I'm asking the question here: If an otherwise English sentence begins with a word that is written in non-Latin script (e.g. Greek letters), does this word have to be capitalized?
There is a similar situation in mathematics, where we have variables. If you start a sentence with a variable, should you capitalize it? Absolutely not, because the variables n and N may mean different things. So should you leave it lowercase? No, because this is inelegant*. So what do you do? The generally accepted solution is never to begin a sentence with a variable. For example, the APA Style Blog says
"Do not begin a sentence with a lowercase statistical term (e.g., t test or p value), a lowercase abbreviation (e.g., lb), or a symbol that stands alone (e.g., α)."
If you have any doubts about whether your readers will know that λογος and Λογος are the same, use this rule. For example, rather than saying
Λογος is a word that ...
The word λογος ...
If you're writing for an audience that you are sure knows enough about the foreign alphabet to recognize capital letters, you can ignore this advice and capitalize foreign words at the beginning of a sentence.
*but much, much, better than capitalizing it.
I.M. Mills and W.V. Metanomski's On the use of italic and roman fonts for symbols in scientific text provides some guidance on the use of Greek characters in scientific uses. All the examples given use lower case.
However, stylistically, this does not necessarily work in other circumstances and I think there is not a rule. Ancient Greek did not have a distinction between upper and lower case, but modern Greek does. Some Ancient Green (and Latin) modern texts use all upper, all lower or English-style sentence casing. Basically - it's a bit all over the place.
I would capitalise it in cases where the reader would be reasonably expected to know Greek letters and write it in (italicised) Latin script where they wouldn't with the Green afterwards in brackets.
Think of your reader - if you expect that they know the Greek alphabet, they probably know upper and lower cases. If not, use the Latin alphabet with Greek as a follow-up.