| L(s) = 1 | − 2.62·2-s + 3-s + 4.89·4-s − 3.97·5-s − 2.62·6-s − 7-s − 7.60·8-s + 9-s + 10.4·10-s + 3.07·11-s + 4.89·12-s − 5.58·13-s + 2.62·14-s − 3.97·15-s + 10.1·16-s − 3.53·17-s − 2.62·18-s + 4.27·19-s − 19.4·20-s − 21-s − 8.06·22-s − 8.48·23-s − 7.60·24-s + 10.8·25-s + 14.6·26-s + 27-s − 4.89·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.44·4-s − 1.77·5-s − 1.07·6-s − 0.377·7-s − 2.68·8-s + 0.333·9-s + 3.30·10-s + 0.926·11-s + 1.41·12-s − 1.54·13-s + 0.701·14-s − 1.02·15-s + 2.54·16-s − 0.856·17-s − 0.618·18-s + 0.980·19-s − 4.35·20-s − 0.218·21-s − 1.72·22-s − 1.76·23-s − 1.55·24-s + 2.16·25-s + 2.87·26-s + 0.192·27-s − 0.925·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 + 5.58T + 13T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 + 0.0532T + 37T^{2} \) |
| 41 | \( 1 - 4.90T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 9.25T + 47T^{2} \) |
| 53 | \( 1 + 2.19T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 0.516T + 97T^{2} \) |
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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
−7.71441116039997099468706465354, −7.13775753828926281106018538050, −6.76953515030288948059954594890, −5.63785206102088062850328638563, −4.26300947381659233775976935602, −3.81655097790387083301237275168, −2.78136574092383950148844559657, −2.13136857509073460583415686690, −0.853091637250061579913274404597, 0,
0.853091637250061579913274404597, 2.13136857509073460583415686690, 2.78136574092383950148844559657, 3.81655097790387083301237275168, 4.26300947381659233775976935602, 5.63785206102088062850328638563, 6.76953515030288948059954594890, 7.13775753828926281106018538050, 7.71441116039997099468706465354
