# RamanujanTauL

gives the Ramanujan tau Dirichlet L-function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• For , is given by the Dirichlet series , where is the Ramanujan function.
• RamanujanTauL can be evaluated to arbitrary numerical precision.
• RamanujanTauL automatically threads over lists.

# Examples

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## Basic Examples(2)

Evaluate numerically:

Plot over a subset of the reals:

## Scope(18)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(2)

Value at zero:

Find a value of x for which RamanujanTauL[x]=0.8:

### Visualization(2)

Plot the RamanujanTauL:

Plot the real part of RamanujanTauL function:

Plot the imaginary part of RamanujanTauL function:

### Function Properties(10)

RamanujanTauL is defined for all real values:

Complex domain:

Bounds on the function range of RamanujanTauL:

RamanujanTauL is an analytic function of x:

RamanujanTauL is neither non-increasing nor non-decreasing:

RamanujanTauL is not injective:

RamanujanTauL is surjective:

RamanujanTauL is neither non-negative nor non-positive:

RamanujanTauL has no singularities or discontinuities:

RamanujanTauL is neither convex nor concave:

## Applications(5)

Plot on the critical line:

Find a zero of RamanujanTauL:

The number of zeros on the critical strip from 0 to :

Plot of the real part:

Define and visualize the Ramanujan function:

## Properties & Relations(5)

Functional equation:

Approximate RamanujanTauL using an Euler product formula:

On the critical line, RamanujanTauL can be split into RamanujanTauTheta and RamanujanTauZ:

Evaluate derivatives numerically:

RamanujanTauZ can be expressed using RamanujanTauL:

Wolfram Research (2007), RamanujanTauL, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauL.html.

#### Text

Wolfram Research (2007), RamanujanTauL, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauL.html.

#### CMS

Wolfram Language. 2007. "RamanujanTauL." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTauL.html.

#### APA

Wolfram Language. (2007). RamanujanTauL. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTauL.html

#### BibTeX

@misc{reference.wolfram_2024_ramanujantaul, author="Wolfram Research", title="{RamanujanTauL}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauL.html}", note=[Accessed: 24-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_ramanujantaul, organization={Wolfram Research}, title={RamanujanTauL}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauL.html}, note=[Accessed: 24-June-2024 ]}