Are papers important when moving to industry?

Hello there!

Currently I am starting my 4th year of postdoc and I am seriously considering moving to industry. I am in a big institution and in a very relevant lab in my field, and I am an author in a couple of papers in CNS journals. The problem is that I don't have any first-author publication during my postdoc and getting it would take me another one or two years. Are papers important when applying to industry jobs, or just showing that I got the experience and expertise is enough?

Thanks!