Most people believe Riemann Hypothesis is true. Since RH has not been proved yet, so it is not completely insane to disprove RH.
Among the ways to disprove RH, straightforward ways, such as: try to find a zero not on critical line, or to find a number n which Mertens function M ( n ) not satisfy "square root error term" condition, those approaches seems not practical. Because we know, even if such a root or a number exist, they will be very big.
What are some practical attempts to disprove Riemann Hypothesis ?
The only 'practical' attempt I know to disprove RH is to prove De Bruijn–Newman constant Λ > 0.
In additional to above attempt, is there any other 'practical' attempt to disprove RH ?