The mass formula gives the total mass as There are two even unimodular lattices of dimension 16, one with root system "E"82 and automorphism group of order 2×6967296002 = 970864271032320000, and one with root system "D"16 and automorphism group of order 21516! = 685597979049984000. So the mass formula is  There are 24 even unimodular lattices of dimension 24, called the Niemeier lattices. The mass formula for them is checked in . The mass in this case is large, more than 40 million. This implies that there are more than 80 million even unimodular lattices of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass. By refining this argument, showed that there are more than a billion such lattices. In higher dimensions the mass, and hence the number of lattices, increases very rapidly. Siegel gave a more general formula that counts the weighted number of representations of one quadratic form by forms in some genus; the Smith–Minkowski–Siegel mass formula is the special case when one form is the zero form. Tamagawa showed that the mass formula was equivalent to the statement that the Tamagawa number of  the orthogonal group is 2, which is equivalent to saying that the Tamagawa number of its simply connected cover the spin group is 1. André Weil conjectured more generally that the Tamagawa number of any simply connected semisimple group is 1, and this conjecture was proved by Kottwitz in 1988.