Help explaining an output of polar plot from Wolfram Alpha

Questions
,
1

In studying complex numbers and calculus in general I have been following a line of logical inquiry to try to normalize the complex plane, while I have now figured out a normalization method I am still stumped as to why wolfram alpha has such a strange response to some other equation I used as an intermediate step. This is the equation in question:

A polar plot of the following:
ζ(-s)^e + Γ^s/(π tan^(-1)(1/sqrt(2)))

The output cycles from one graph, generally to another before stopping and each time there’s some random probability which graph I receive. I’m not sure if it’s random or not, truly, to be clear. Just that I can’t identify any pattern to which result will be displayed.

Included is a sample of one of the graphs it will display and a screenshot of the equation as well as a plain language version:

OutputOne.png
OutputOne.png829×618 49.8 KB

polar plot | ζ(-s)^e + Γ^s/(π tan^(-1)(1/sqrt(2)))

EquationStrange.png
EquationStrange.png880×402 26.1 KB

2

What is Gamma?

3
4

You are writing Gamma^s. Is that the same as the Gamma function calculated at s?

5

Edited : I misunderstood your question. The answer is no. See other comment.

6

Try typing Gamma(s).

7

That would fundamentally change how it’s implemented. I put it there specifically.

Sorry, I see why you’re asking me what gamma IS now. Your question, to me, was ambiguous. “Gamma is gamma.” shrug

So, in this scenario I’m extending the gamma function by our zeta defined s.

8

In this case what’s going on is basically just a cancelling of the factors. (-s) for (s)

9

Thus I projected the following instead:

This again isn’t the final product but is nevertheless descriptive.

polar plot | ζ(-s)^e + Γ ^-1/2*(π tan^(-1)(1/sqrt(2)))