In an interesting paper, Iaroslav Blagouchine has presented, among other things, the equivalence of the explicit analytic continuation of due to Hasse (in this 1930 paper) with one derived 4 years earlier by a Joseph Ser (in this paper) .
It may be that these could help improve the situation regarding the location of the non-trivial zeros of , although of course the classical Mertens approach with logs and the trigonometric identity cannot be mimicked here due to the ever growing number of terms added for each new .
But who was Joseph Ser then ? There’s a short wikipedia bio saying that nothing much is known about him. There seems to be a genealogical entry for him here, with little more details beyond dates. I couldn’t find his name among Normaliens nor Polytechniciens. He also doesn’t appear in the list of pre-1901 professeurs agrégés, and neither in the list of PhDs compiled by Hélène Gispert in her book.
As for published material, Numdam has 3 papers from him, where only the earliest one has a tiny indication: Nantes. One can also find that a book of Ser, Les Calculs Formels des Séries de Factorielles, got a quite unfavorable review in Bull AMS, while the same book got a more positive review in L’Enseignement Mathématique. And finally, he seems to have authored several articles in Mathesis, at least until 1953 when he was well into his seventies.
Feel free to comment if you know more about his work and career.