OFFSET
1,1
COMMENTS
As discussed in A257819, the real part of li(z) is a well behaved function for any real z, except for the singularity at z=+1. It has three roots: z=A070769 (the Soldner's constant), z=0, and z=-a. However, unlike in the other two cases, the imaginary part of li(-a) is not infinitesimal in the neighborhood of this root; it is described in A257822.
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function
FORMULA
Satisfies real(li(-a)) = 0.
EXAMPLE
2.4664082624126780751971033507759329502907808782774099823786...
MATHEMATICA
RealDigits[a/.FindRoot[Re[LogIntegral[-a]]==0, {a, 2}, WorkingPrecision->120]][[1]] (* Vaclav Kotesovec, May 11 2015 *)
PROG
(PARI) li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
return(+Pi*I-eint1(-log(c)))); }
a=-solve(x=-3, -1, real(li(x))) \\ Better use excess realprecision
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, May 11 2015
STATUS
approved