We show that the resulting manifold by $r$-surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope $r$ satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates $L$-spaces and the left-orderability of their fundamental groups.
"On left-orderable fundamental groups and Dehn surgeries on knots." J. Math. Soc. Japan 67 (1) 319 - 338, January, 2015. https://doi.org/10.2969/jmsj/06710319