It is also known that the nontrivial zeros are symmetrically placed about the , a result which follows from the functional equation and the symmetry about the line . For if is a nontrivial zero, then is also a zero (by the functional equation), and then is another zero. But and are symmetrically placed about the line , since , and if , then . The Riemann hypothesis is equivalent to , where is the (Csordas 1994). It is also equivalent to the assertion that for some constant ,
André Weil proved the Riemann hypothesis to be true for field functions (Weil 1948, Eichler 1966, Ball and Coxeter 1987). In 1974, Levinson (1974ab) showed that at least 1/3 of the must lie on the (Le Lionnais 1983), a result which has since been sharpened to 40% (Vardi 1991, p. 142). It is known that the zeros are symmetrically placed about the line . This follows from the fact that, for all complex numbers ,
It is well established that composite faces manufactured using these computer graphic methods tend to be perceived as more attractive than the average attractiveness rating of their constituent images (as Galton's composites also were). That composite faces tend to be judged as more attractive than their constituent images led many researchers to conclude that ‘attractive faces are only average’. In other words, many researchers that had noted the high attractiveness of composite faces proposed that averageness is the critical determinant of attractiveness. This is often referred to as the ‘Averageness Hypothesis’. One explanation put forward for the effect of averageness on facial attractiveness is that average faces most closely resemble mental representations of a typical face and can therefore be processed most easily by the visual system (this explanation is similar to the perceptual bias account of symmetry preferences in that it emphasises a possible link between the ease with which faces can be processed and their attractiveness).