  We construct new invariant polynomial for long virtual knots. It is a
generalization of Alexander polynomial. We designate it by $\zeta$ meaning an
analogy with $\zeta$-polynomial for virtual links. A degree of
$\zeta$-polynomial estimates a virtual crossing number. We describe some
application of $\zeta$-polynomial for the study of minimal long virtual
diagrams with respect number of virtual crossings.
