I don't know how good the bound is you can obtain from this, but what about taking a Korkine-Zolotarev reduced basis of Lambda, say (b1,dots,bn): then, by this paper, |bi|22lefraci+34lambdai(Lambda)2, where lambdai(Lambda) is the i-th successive minimum of Lambda. By Minkowski, prodni=1lambdai(Lambda)legamman/2ndetLambda=gamman/2n (in your case), gamman being the n-th Hermite constant, whence you get Aleprodni=1|bi|2lefracgamman/2n2nprodni=1sqrti+3.
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