From the paper:<p>&quot;Another noteworthy feature of our work is that we need not restrict ourselves to the leading
coefficient in the Taylor expansion of the L-functions: our formula is about the r-th Taylor
coefficient of the L-function regardless whether r is the central vanishing order or not. This
leads us to speculate that, contrary to the usual belief, central derivatives of arbitrary order of
motivic L-functions (for instance, those associated to elliptic curves) should bear some geometric
meaning in the number field case.&quot;<p>A one-line Clay Prize motivation: Geometric interpretations of higher order derivatives of L-functions could potentially be leveraged (or more likely, illuminate a path) to make progress on conjectures about order of vanishing of L-functions, e.g. Birch Swinnerton-Dyer.