I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)
See, for example, P. Sebah, X. Gourdon, Introduction to the Gamma Function, available here.
Topic 5.1.5, page 13, is about Zeros of the digamma function. We can see that on the negative axis, the digamma function has a single zero between each consecutive negative integers (the poles of the gamma function).
The authors presents the first five zeros of the digamma function on the negative axis with 50 decimal places.
I have found this beautiful manuscript written by Hermite, with reference to gamma functions, Cours de M. Hermite, Librairie Scientifique A. Hermann, 1883.
Also, the complete Oeuvres de Charles Hermite is available here.
Another reference is NIST Digital Library of Mathematical Functions