Yves explained to me the following idea, which he attributes to Sunseri.
In his project, he explicitly computes a power series a_0+a_1T+a_2T^2+...
representing the "component" of the 37-adic zeta function with the zero.
By "explicitly computes", he means that he considered the first 1000 or so
a_i, and computed the first 1000 terms of each of their 37-adic
expansions. He then computes the zero using Hensel's lemma. However, he
says that if he had not bothered computing the a_i explicitly, and just
used the formula for L_p(chi,s) in Washington's book on Cyclotomic
Fields to evaluate L_p(T) (and its derivative) for certain *explicit* T
in Z_p, he would still have had enough data to use Hensel's lemma to
find the root, and that the computation would have been an order of
magnitude faster. Hence he estimates that one could easily compute
zeros in this way, to thousands and thousands of significant figures,
nowadays (Aug 2001). Yves tells me that this is what Sunseri did
originally, but his work was not published either.