Let Lambda be the class of nonnegative sequences (lambda(n)) increasing to +infinity, A is an element of(-infinity, +infinity], L-A be the class of continuous functions increasing to +infinity on [A(0), A), (lambda(n)) is an element of Lambda, and F (s) = Sigma a(n)e(s lambda n) be a Dirichlet series such that its maximum term mu(sigma, F) = max(n) vertical bar a(n)vertical bar e(sigma lambda n) is defined for every sigma is an element of (-infinity, A). It is proved that for all functions alpha is an element of L+infinity and beta is an element of L-A the equality rho(*)(alpha,beta)(F) = max((eta n)is an element of Lambda) (lim) over bar (n ->infinity) alpha(eta(n))/beta(eta(n)/lambda(n) + 1/lambda(n) ln 1/vertical bar a(n)vertical bar holds, where rho(*)(alpha,beta)(F) is the generalized alpha,beta-order of the function ln mu(sigma, F), i.e. rho(alpha,)beta(*)(F) = 0 if the function mu(sigma, Gamma) is bounded on (-infinity, A), and rho(*)(alpha,beta)(F) = (lim) over bar (sigma up arrow A) alpha(ln mu(sigma, Gamma)) / beta(sigma) if the function sigma(sigma, F) is unbounded on (-infinity, A).
