Period polynomials have long been fruitful tools for the study of values of $L$-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways
to incorporate non-cuspidal modular forms and values of derivatives of $L$-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for $L$-derivatives.
In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state gen
eral conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative period polynomials in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series.
