L(s) = 1 | + i·5-s + 1.15i·7-s + (1.08 + 3.13i)11-s + 6.48i·13-s − 0.176·17-s + 5.77i·19-s + 7.84i·23-s − 25-s + 9.47·29-s + 0.163·31-s − 1.15·35-s + 3.01·37-s + 2.86·41-s − 5.38i·43-s − 2.23i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.436i·7-s + (0.326 + 0.945i)11-s + 1.79i·13-s − 0.0427·17-s + 1.32i·19-s + 1.63i·23-s − 0.200·25-s + 1.75·29-s + 0.0293·31-s − 0.195·35-s + 0.496·37-s + 0.447·41-s − 0.820i·43-s − 0.325i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.982366728\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.982366728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + (-1.08 - 3.13i)T \) |
good | 7 | \( 1 - 1.15iT - 7T^{2} \) |
| 13 | \( 1 - 6.48iT - 13T^{2} \) |
| 17 | \( 1 + 0.176T + 17T^{2} \) |
| 19 | \( 1 - 5.77iT - 19T^{2} \) |
| 23 | \( 1 - 7.84iT - 23T^{2} \) |
| 29 | \( 1 - 9.47T + 29T^{2} \) |
| 31 | \( 1 - 0.163T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 2.86T + 41T^{2} \) |
| 43 | \( 1 + 5.38iT - 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 + 3.01iT - 53T^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 8.92iT - 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 - 8.28iT - 71T^{2} \) |
| 73 | \( 1 + 16.0iT - 73T^{2} \) |
| 79 | \( 1 + 6.76iT - 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + 11.0iT - 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.021828091881970479148129417726, −7.32538429549755070022854542869, −6.79751176509586541784609920215, −6.10562085811350313208293094322, −5.42405888425601409802668823965, −4.38357424665067872752956260812, −4.01364431502218433615711600368, −2.97549784767547127312552191053, −2.03722267107118002646569408940, −1.42988248827003305352967862106,
0.57270086355851070776647180696, 0.947543367187198301093991789293, 2.59170841222921971489920792165, 3.02764007461777971600496126007, 4.06315748974048470848639844074, 4.78389191530063879635277789917, 5.39122495403140975141562911279, 6.30731853776420475380776905486, 6.69754108082986071378381110092, 7.80232777379590012183895281520