I have pH (6.5ish), [Ca] (~400 mg/l) and alkalinity (~360 mg/l), but when I try to do this with the formula that was given to me I get nonsense results.
The saturation index for calcite may be calculated with different reactions. Two of them use HCO3- concentration, the other one uses CO32-. Your alcalinity value may be equal, normally, to HCO3-, but you must check that you use the good equation for the saturation index.
With a pH of 6.5 your solution is acidic (< 7.0) and so saturation is impossible.
I assume that your solution is a dilute solution. Convert all your concentrations in molar units. Calculate the carbonate ion concentration from the alkalinity using the second dissociation constant of carbonic acid (4.68 x 10^-11) and the pH (you should get ~0.9 uM). Multiply this result by the calcium concentration (~10 mM) and divide by the calcite solubility solubility constant (10^-8.48) to get the saturation state. The saturation index is the logarithm of the saturation state. This calculation assumes that you have an ideal solution, i.e., ion activity coefficients are equal to one. Otherwise, use PHREEQC online ((http://www.ndsu.nodak.edu/webphreeq/), it will even take concentrations in mg/L as input.
How about trying CO2sys for this kind of calculations:
http://cdiac.ess-dive.lbl.gov/ftp/co2sys/?C=S;O=A
It might at least help you to calculate [CO32-] from pH and alkalinity.
Langelier Saturation Index Calculator is here:
https://www.lenntech.com/calculators/langelier/index/langelier.htm
It is also necessary to know the temperature and concentration of HCO3- in mg / l
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