| L(s) = 1 | − 2.17·2-s + 3-s + 2.73·4-s − 0.898·5-s − 2.17·6-s + 0.611·7-s − 1.60·8-s + 9-s + 1.95·10-s + 0.653·11-s + 2.73·12-s + 13-s − 1.33·14-s − 0.898·15-s − 1.98·16-s − 0.651·17-s − 2.17·18-s − 5.64·19-s − 2.46·20-s + 0.611·21-s − 1.42·22-s + 1.83·23-s − 1.60·24-s − 4.19·25-s − 2.17·26-s + 27-s + 1.67·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 0.577·3-s + 1.36·4-s − 0.402·5-s − 0.888·6-s + 0.231·7-s − 0.567·8-s + 0.333·9-s + 0.618·10-s + 0.197·11-s + 0.790·12-s + 0.277·13-s − 0.355·14-s − 0.232·15-s − 0.495·16-s − 0.158·17-s − 0.513·18-s − 1.29·19-s − 0.550·20-s + 0.133·21-s − 0.303·22-s + 0.382·23-s − 0.327·24-s − 0.838·25-s − 0.426·26-s + 0.192·27-s + 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 + 0.898T + 5T^{2} \) |
| 7 | \( 1 - 0.611T + 7T^{2} \) |
| 11 | \( 1 - 0.653T + 11T^{2} \) |
| 17 | \( 1 + 0.651T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 9.63T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 + 9.96T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 - 4.50T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 3.66T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−8.260269456834465559424435985403, −7.73366109996220033067733501701, −6.77115369179883197033854918689, −6.40966318696360996376427603844, −4.94852588420328727553414566837, −4.19272082589902939200612215230, −3.15483046466521747345589977355, −2.13809534587981644869650086216, −1.31297403068176405174585316384, 0,
1.31297403068176405174585316384, 2.13809534587981644869650086216, 3.15483046466521747345589977355, 4.19272082589902939200612215230, 4.94852588420328727553414566837, 6.40966318696360996376427603844, 6.77115369179883197033854918689, 7.73366109996220033067733501701, 8.260269456834465559424435985403
