I've been playing around with domain coloring recently, and discovered a bug in PARI's lngamma function-- it would give results off by 2πi when the real part of the input was exactly -1/2. (I was graphing the Z function at the time.)

It turns out the problem had to do with the use of Euler's reflection formula, which necessitates the calculation of log sin(πz) with branch cuts along (-∞,0] and [1,∞). In principle we know exactly what integer multiple of 2πi to add to the principal value of the logarithm, but unfortunately we can't rely on this prediction because of numerical instabilities in the computation of sin(πz). (Actually, PARI is so frakking brilliant that this problem really only shows up when Re(z) is exactly a half-integer.) So the solution is of course to compute the logarithm first, see what imaginary part we get, and then add the appropriate multiple of 2πi.

And of course I had to make some domain coloring plots to check that I'd fixed things up. :^) The pictures are of the region [-10,10] x [-10,10]. (scroll right to see a high-contrast diff of the two)

Original:	Fixed:	Diff: