A Gaussian filter is a filter whose impulse response is a Gaussian function. Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.  Wikipedia
See the online filter calculators and plotters here.
Here are the lowpass prototype poles for a Gaussian filter to 6 dB and to 12 dB. Lowpass prototype inductor and capacitor values are here.


Data taken from Analog Devices app note.