| L(s) = 1 | − 2.37·2-s − 2.00·3-s + 3.64·4-s + 0.380·5-s + 4.76·6-s + 3.69·7-s − 3.89·8-s + 1.02·9-s − 0.904·10-s − 5.88·11-s − 7.30·12-s + 3.92·13-s − 8.77·14-s − 0.763·15-s + 1.97·16-s + 7.17·17-s − 2.43·18-s − 2.89·19-s + 1.38·20-s − 7.41·21-s + 13.9·22-s − 2.31·23-s + 7.81·24-s − 4.85·25-s − 9.32·26-s + 3.96·27-s + 13.4·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 1.15·3-s + 1.82·4-s + 0.170·5-s + 1.94·6-s + 1.39·7-s − 1.37·8-s + 0.341·9-s − 0.285·10-s − 1.77·11-s − 2.10·12-s + 1.08·13-s − 2.34·14-s − 0.197·15-s + 0.493·16-s + 1.74·17-s − 0.572·18-s − 0.663·19-s + 0.309·20-s − 1.61·21-s + 2.98·22-s − 0.482·23-s + 1.59·24-s − 0.971·25-s − 1.82·26-s + 0.763·27-s + 2.54·28-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 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 - 0.380T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 + 5.88T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + 2.31T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 0.380T + 43T^{2} \) |
| 47 | \( 1 + 0.614T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 3.52T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 0.687T + 79T^{2} \) |
| 83 | \( 1 - 1.33T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 6.95T + 97T^{2} \) |
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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
−8.147146645519799350112284236808, −7.79483215396094716054150402617, −6.72298643013731316066657448921, −5.92906213454967297944535373192, −5.33716742797076687788679440012, −4.59279349774268872414685617791, −3.05057205328735081602403181050, −1.89612030042986217935446071472, −1.12741168140082976611640941575, 0,
1.12741168140082976611640941575, 1.89612030042986217935446071472, 3.05057205328735081602403181050, 4.59279349774268872414685617791, 5.33716742797076687788679440012, 5.92906213454967297944535373192, 6.72298643013731316066657448921, 7.79483215396094716054150402617, 8.147146645519799350112284236808
