L(s) = 1 | + 1.06·2-s + 0.817·3-s − 0.864·4-s − 5-s + 0.870·6-s + 0.699·7-s − 3.05·8-s − 2.33·9-s − 1.06·10-s + 11-s − 0.706·12-s + 2.77·13-s + 0.745·14-s − 0.817·15-s − 1.52·16-s − 0.215·17-s − 2.48·18-s + 2.38·19-s + 0.864·20-s + 0.571·21-s + 1.06·22-s + 3.64·23-s − 2.49·24-s + 25-s + 2.95·26-s − 4.35·27-s − 0.604·28-s + ⋯ |
L(s) = 1 | + 0.753·2-s + 0.471·3-s − 0.432·4-s − 0.447·5-s + 0.355·6-s + 0.264·7-s − 1.07·8-s − 0.777·9-s − 0.336·10-s + 0.301·11-s − 0.203·12-s + 0.769·13-s + 0.199·14-s − 0.211·15-s − 0.380·16-s − 0.0522·17-s − 0.585·18-s + 0.547·19-s + 0.193·20-s + 0.124·21-s + 0.227·22-s + 0.759·23-s − 0.509·24-s + 0.200·25-s + 0.579·26-s − 0.838·27-s − 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 3 | \( 1 - 0.817T + 3T^{2} \) |
| 7 | \( 1 - 0.699T + 7T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 + 0.215T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.667T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 9.09T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 + 8.26T + 83T^{2} \) |
| 89 | \( 1 + 9.11T + 89T^{2} \) |
| 97 | \( 1 + 9.33T + 97T^{2} \) |
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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
−8.303263942080299514954898223805, −7.36742580811471630506939075724, −6.47950999266234092182675711558, −5.66071048349602274305042848203, −5.04404058568613350075957368609, −4.19312253358990626458604831518, −3.39608128787997914656456776038, −2.95894192141611323672236533126, −1.51435499869840016292537832244, 0,
1.51435499869840016292537832244, 2.95894192141611323672236533126, 3.39608128787997914656456776038, 4.19312253358990626458604831518, 5.04404058568613350075957368609, 5.66071048349602274305042848203, 6.47950999266234092182675711558, 7.36742580811471630506939075724, 8.303263942080299514954898223805