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A
 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
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
G
 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
I
 1, 2, 3, 4, 5, 6, 7, 8, 9
J
 1, 2, 3, 4, 5, 6, 7, 8, 9

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

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

Flower (nice and easy)

This puzzle contains no zero.

Across

A across = G across - L across
E across = I across + 1
F across = reverse(K down) + K across (this is redundant information as it follows from the other cues)
G across = H down ^ 2
Prime number(I across) (redundant)
K across = K down - 1 (redundant)
L across = E across ^ 3

Down
Palindrome(A down)
Prime number(B down) (redundant)
C down = H down + J down - 1 (redundant)
D down = C down * reverse(K down)
reverse(J down) = K down + digit product(K across) (redundant)
square number(K down) (redundant)