The mechanisms behind the threshold-voltage shift in organic transistors due to functionalizing of the gate dielectric with self-assembled monolayers (SAMs) are still under debate. We address the mechanisms by which SAMs determine the threshold voltage, by analyzing whether the threshold voltage depends on the gate-dielectric capacitance. We have investigated transistors based on five oxide thicknesses and two SAMs with rather diverse chemical properties, using the benchmark organic semiconductor dinaphtho[2,3-b:2′,3′-f]thieno[3,2-b]thiophene. Unlike several previous studies, we have found that the dependence of the threshold voltage on the gate-dielectric capacitance is completely different for the two SAMs. In transistors with an alkyl SAM, the threshold voltage does not depend on the gate-dielectric capacitance and is determined mainly by the dipolar character of the SAM, whereas in transistors with a fluoroalkyl SAM the threshold voltages exhibit a linear dependence on the inverse of the gate-dielectric capacitance. Kelvin probe force microscopy measurements indicate this behavior is attributed to an electronic coupling between the fluoroalkyl SAM and the organic semiconductor.