Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere.

Let GGLn be a semisimple linear algebraic group over Q. The Tamagawa measure μ on the group of adelic points G(A) is uniquely determined by the product formula.
Nevertheless, it can be written as a product of local factors μ=p μp in a non-unique way.
For each place p< choose a special maximal compact open subgroup KpG(Qp) in a way that at almost all places these subgroups are hyper-special and coincide with G(Zp)=G(Qp)GLn(Zp). Then normalise the local measures μp so as to satisfy μp(KP)=1 for p<. For p= choose m to be the measure which comes from the canonical Riemannian metric on the symmetric space X=G(R)/K, where K is a maximal compact subgroup of G(R), where the metric is given by the Killing form.
Then μ=cm for some c>0 and I would like to know this number c.

The Kp are not unique which results in c not being unique up to a rational number and I think not all rational numbers are possible. I expect that c is a rational number times a power of π and it would already help to know this power.

If no general result is known, it would help to have these numbers in special cases like SLn, PGLn, Sp2n and so on.

Answer

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