| L(s) = 1 | − 3.82·2-s + 3.57·3-s + 6.64·4-s + 6.65·5-s − 13.6·6-s + 1.73·7-s + 5.19·8-s − 14.1·9-s − 25.4·10-s + 50.6·11-s + 23.7·12-s − 60.3·13-s − 6.63·14-s + 23.7·15-s − 73.0·16-s + 26.6·17-s + 54.3·18-s − 54.2·19-s + 44.1·20-s + 6.20·21-s − 193.·22-s + 23·23-s + 18.5·24-s − 80.7·25-s + 231.·26-s − 147.·27-s + 11.5·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.688·3-s + 0.830·4-s + 0.594·5-s − 0.931·6-s + 0.0935·7-s + 0.229·8-s − 0.525·9-s − 0.804·10-s + 1.38·11-s + 0.571·12-s − 1.28·13-s − 0.126·14-s + 0.409·15-s − 1.14·16-s + 0.380·17-s + 0.711·18-s − 0.654·19-s + 0.493·20-s + 0.0644·21-s − 1.87·22-s + 0.208·23-s + 0.158·24-s − 0.646·25-s + 1.74·26-s − 1.05·27-s + 0.0777·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 3 | \( 1 - 3.57T + 27T^{2} \) |
| 5 | \( 1 - 6.65T + 125T^{2} \) |
| 7 | \( 1 - 1.73T + 343T^{2} \) |
| 11 | \( 1 - 50.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.2T + 6.85e3T^{2} \) |
| 31 | \( 1 + 98.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 92.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 323.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 515.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 685.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 88.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 18.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 751.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 604.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 556.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 262.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−9.672552907243153120844546719030, −8.856587021981692667101997345865, −8.253694914613065662980785146001, −7.30456517749033037010822138775, −6.45750741942397384086462495279, −5.17212017950641414445405198410, −3.83160615221313631895199908653, −2.41353423969828427075370111212, −1.54241029209857219536918048432, 0,
1.54241029209857219536918048432, 2.41353423969828427075370111212, 3.83160615221313631895199908653, 5.17212017950641414445405198410, 6.45750741942397384086462495279, 7.30456517749033037010822138775, 8.253694914613065662980785146001, 8.856587021981692667101997345865, 9.672552907243153120844546719030
