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I took just Arduino Uno board, connected Ethernet Shield to it and compiled Web Server example. And it works.

But it outputs:

analog input 0 is 1023
analog input 1 is 1023
analog input 2 is 789
analog input 3 is 663
analog input 4 is 525
analog input 5 is 449

Why so different values? Shouldn't it be zeros? Is this static electricity or what?

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Floating inputs, EMI, high input impedance and sample-and-hold capacitor. It can measure anything at this point (usually electro static charge is split between pin capacity and S&H capacitor. It can be used even for touch sense pins...

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