| L(s) = 1 | − 2.81·2-s + 1.48·3-s + 5.93·4-s + 2.03·5-s − 4.18·6-s − 2.55·7-s − 11.0·8-s − 0.790·9-s − 5.72·10-s + 3.01·11-s + 8.81·12-s + 2.19·13-s + 7.20·14-s + 3.01·15-s + 19.3·16-s − 4.87·17-s + 2.22·18-s + 3.05·19-s + 12.0·20-s − 3.80·21-s − 8.49·22-s − 2.52·23-s − 16.4·24-s − 0.872·25-s − 6.16·26-s − 5.63·27-s − 15.1·28-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 0.858·3-s + 2.96·4-s + 0.908·5-s − 1.70·6-s − 0.967·7-s − 3.91·8-s − 0.263·9-s − 1.80·10-s + 0.909·11-s + 2.54·12-s + 0.607·13-s + 1.92·14-s + 0.779·15-s + 4.82·16-s − 1.18·17-s + 0.525·18-s + 0.700·19-s + 2.69·20-s − 0.830·21-s − 1.81·22-s − 0.526·23-s − 3.35·24-s − 0.174·25-s − 1.20·26-s − 1.08·27-s − 2.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 2.03T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 - 7.43T + 41T^{2} \) |
| 43 | \( 1 - 3.81T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.272454910811058224958363667742, −7.67391964592328054908662630919, −6.63354318869949195845101829147, −6.38871397149378578649146904405, −5.62125761784765521829799247649, −3.69274470847428727825822208774, −2.99200062905803248966152933990, −2.15859295596880115870206059661, −1.44580750762839797328006626620, 0,
1.44580750762839797328006626620, 2.15859295596880115870206059661, 2.99200062905803248966152933990, 3.69274470847428727825822208774, 5.62125761784765521829799247649, 6.38871397149378578649146904405, 6.63354318869949195845101829147, 7.67391964592328054908662630919, 8.272454910811058224958363667742
