For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided the X_i are not too far apart in logarithmic size.
A counterexample to Malle’s conjecture on the asymptotics of dis- criminants
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Counting $D_4$-field extensions by multi-invariants
For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided the X_i are not too far apart in logarithmic size.