If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few zeros of its zeta function on the critical line?
I tried in pari but pari seems to choke on zetakinit.
I tried in magma and got much further. I can create the number field, and use the LSeries command to compute some form of the $L$-function. I can now evaluate the $L$-function at pretty much any point I want on the critical line, and use things like LSetPrecision to warn magma that I'm going up the critical line. I have no feeling for these things though; I don't even know how far I might expect to look up the line for the first, say, five zeros. The main problem I have though is that I'm just naively evaluating the function at some random points, and each evaluation might take a minute, and I evaluate the function at a point and it's non-zero and now I don't even know whether to move up or down.
Are there any other computer algebra packages that might be able to help me out?