Follow-up Comment #2, bug #50343 (project gsl):

> What are the values returned by MathieuCharacteristicA[0, -1] 

You can check this with Wolfram|Alpha (which is powered by Mathematica):
http://www.wolframalpha.com/input/?i=MathieuCharacteristicA%5B0,+-1%5D

The result is

-0.45513860410741354823263318752888586691651984219666547103...

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