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Abstract. We show how to calculate the zeta functions and the orders |X| of Tate-Shafarevich groups of the elliptic curves with equation Y 2 + XY = X 3 + αX 2 + const · T −k over the rational function field Fq(T ), where q is a power of 2. In the range q = 2, k ≤ 37, α ∈ F 2 [T −1 ] odd of degree ≤ 19, the largest values obtained for |X| are 47 2 (one case), 39 2 (one case) and 27 2 (three cases).We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands correspondence for GL(2) over local or global fields of characteristic two.