For non-Kummer cubic L-functions over F_q[t], the mixed shifted moments are at most |C_g| g^{k_1^2+...+k_m^2} times pairwise shift factors, matching the conjectured unitary order.
Chandee, On the correlation of shifted values of the Riemann zeta function, The Quarterly Journal of Mathematics , 62 (2011), pp
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Upper bound for the moment of shifted values of cubic $L$-functions over function fields
For non-Kummer cubic L-functions over F_q[t], the mixed shifted moments are at most |C_g| g^{k_1^2+...+k_m^2} times pairwise shift factors, matching the conjectured unitary order.