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A
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9
B
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9
C
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9
D
 1, 2, 3, 4, 5, 6, 7, 8, 9
E
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9
F
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9
G
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9
H
 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

 1, 2, 3, 4, 5, 6, 7, 8, 9

Face

No zeros occur in this puzzle.
Across:
A: The reverse of F down multiplied with a number that is both prime and palindromic. Also, A across has the same digit product as B down.
D: A down + C down
G: All digits of G across are the same.
H: Subtract the digit product of B down from G across and multiply the result with G across to get H across.
Down:
A: This palindrome equals B down multiplied with E down.
B: This palindrome equals E down multiplied with G across.
C: The digit sum of C down is a cubic number.
E: A prime number.
F: A prime number.