The principal tools for computing are the expansion (25.2.9) for general values of , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for . Details are provided in Haselgrove and Miller (1960). See also Allasia and Besenghi (1989), Butzer and Hauss (1992), Kerimov (1980), and Yeremin et al. (1985). Calculations relating to derivatives of and/or can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988).
Most numerical calculations of the Riemann zeta function are concerned with locating zeros of in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of lie on the critical line . Calculations to date (2008) have found no nontrivial zeros off the critical line. For recent investigations see, for example, van de Lune et al. (1986) and Odlyzko (1987). For earlier work see Haselgrove and Miller (1960).