L(s) = 1 | − 2-s + 0.722·3-s + 4-s + 5-s − 0.722·6-s + 3.72·7-s − 8-s − 2.47·9-s − 10-s + 11-s + 0.722·12-s − 4.31·13-s − 3.72·14-s + 0.722·15-s + 16-s − 2.90·17-s + 2.47·18-s + 6.16·19-s + 20-s + 2.69·21-s − 22-s − 4.98·23-s − 0.722·24-s + 25-s + 4.31·26-s − 3.96·27-s + 3.72·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.417·3-s + 0.5·4-s + 0.447·5-s − 0.295·6-s + 1.40·7-s − 0.353·8-s − 0.825·9-s − 0.316·10-s + 0.301·11-s + 0.208·12-s − 1.19·13-s − 0.995·14-s + 0.186·15-s + 0.250·16-s − 0.705·17-s + 0.583·18-s + 1.41·19-s + 0.223·20-s + 0.587·21-s − 0.213·22-s − 1.03·23-s − 0.147·24-s + 0.200·25-s + 0.845·26-s − 0.762·27-s + 0.703·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 - 0.722T + 3T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 - 6.16T + 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 0.457T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 - 5.08T + 59T^{2} \) |
| 61 | \( 1 + 4.67T + 61T^{2} \) |
| 67 | \( 1 + 9.57T + 67T^{2} \) |
| 71 | \( 1 + 6.97T + 71T^{2} \) |
| 79 | \( 1 + 2.97T + 79T^{2} \) |
| 83 | \( 1 - 8.54T + 83T^{2} \) |
| 89 | \( 1 - 8.38T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−7.61397003167406303736249689453, −7.07561554053386270378033234341, −6.15459740139072273310985916158, −5.29376176227050471015887965503, −4.91277438301390281031200766305, −3.78244245374932126425600141306, −2.81173493334253370353279342739, −2.08789789956527561438535517042, −1.43812417456871051370451042862, 0,
1.43812417456871051370451042862, 2.08789789956527561438535517042, 2.81173493334253370353279342739, 3.78244245374932126425600141306, 4.91277438301390281031200766305, 5.29376176227050471015887965503, 6.15459740139072273310985916158, 7.07561554053386270378033234341, 7.61397003167406303736249689453