| L(s) = 1 | − 2-s − 1.92·3-s + 4-s − 0.749·5-s + 1.92·6-s − 4.27·7-s − 8-s + 0.691·9-s + 0.749·10-s + 1.34·11-s − 1.92·12-s + 6.84·13-s + 4.27·14-s + 1.43·15-s + 16-s − 5.15·17-s − 0.691·18-s − 2.23·19-s − 0.749·20-s + 8.20·21-s − 1.34·22-s + 2.73·23-s + 1.92·24-s − 4.43·25-s − 6.84·26-s + 4.43·27-s − 4.27·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s − 0.335·5-s + 0.784·6-s − 1.61·7-s − 0.353·8-s + 0.230·9-s + 0.236·10-s + 0.405·11-s − 0.554·12-s + 1.89·13-s + 1.14·14-s + 0.371·15-s + 0.250·16-s − 1.25·17-s − 0.162·18-s − 0.513·19-s − 0.167·20-s + 1.79·21-s − 0.286·22-s + 0.570·23-s + 0.392·24-s − 0.887·25-s − 1.34·26-s + 0.853·27-s − 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.92T + 3T^{2} \) |
| 5 | \( 1 + 0.749T + 5T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 - 6.84T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 0.0238T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.312825878842172149986570171741, −6.95251298266109535668374930639, −6.69462433566022200100142096702, −6.04423699032952110666075873922, −5.45764008012120686013375719781, −4.02335178128463992259734801060, −3.56650197225111238258038880267, −2.33869435519210508374742345269, −0.931214111945643139819119879388, 0,
0.931214111945643139819119879388, 2.33869435519210508374742345269, 3.56650197225111238258038880267, 4.02335178128463992259734801060, 5.45764008012120686013375719781, 6.04423699032952110666075873922, 6.69462433566022200100142096702, 6.95251298266109535668374930639, 8.312825878842172149986570171741
