What is the purpose of 10nH inductance before 1nF(1000pF) bypass capacitor at 2GHz?

Filtering?

I suspect you are not describing the circuit correctly. post a picture or a detailed schematic

Can be required by design such as a part of a DC bias circuit but most likely is it a way to reduce ESD problems.

Here is schematic:
http://www.eta.chalmers.se/~upda/quadrupler.html

the fet gate needs a slightly negative bias voltage on the gate to operate correctly. So they use a bias network to inject DC voltage onto the gate, without adding any RF insertion loss to the RF path. The 10 NH and 100 ohm resistor act as an "open circuit" to the RF signal, and the 1000 pf shunt cap keeps any stray RF from leaking back to the power supply.

The circuit is wrong as drawn, there needs to be a series DC blocking capacitor at the RF input port. Without it, the amp will probably not work. at 2 GHz, a 100 pF series cap should be sufficient.



the fet gate needs a slightly negative bias voltage on the gate to operate correctly. So they use a bias network to inject DC voltage onto the gate, without adding any RF insertion loss to the RF path. The 10 NH and 100 ohm resistor act as an "open circuit" to the RF signal, and the 1000 pf shunt cap keeps any stray RF from leaking back to the power supply.

The circuit is wrong as-drawn, there needs to be a series DC blocking capacitor at the RF input port (labeled "2.5-3 ghz"). Without it, the amp will probably not work. at 2 GHz, a 100 pF series cap should be sufficient.

How to choose such inductor for other frequency? Or, can i use other elements in this scheme?

Impedance increases with increasing frequency. At 2GHz with 10nH inductor we have:

F=2*10^9F = 2000000000

L=10*10^-9L = 0.00000001

X=2*Pi*F*LX = 125.663706144

I agree with BIFF44 concerning the schematic. The coil in series with FET gate may be needed for noise matching. In other LNAs I saw parallel coils, at 3 GHz only a half-turn to "GND", for the same purpose. Such inductor should be adjusted for the lowest noise figure during testing.

Your calculation is correct, but I don't expect that the RL circuit is intended as matching network. It's more likely meaned as pure bias network, as biff44 suggested. Although you may want to put in a higher inductance value, it's not easy to make it work at 3 GHz. So the simple reason might be, that the original designer didn't have higher valued good microwave inductors.

You surely can't use "any other inductor" or "some resistor". The parts must have suitable microwave properties. Particularly the inductor's SRF should not be below the maximum operating frequency.