| L(s) = 1 | − 3.67·2-s + 10.1·3-s + 5.50·4-s − 4.25·5-s − 37.4·6-s − 16.2·7-s + 9.15·8-s + 76.6·9-s + 15.6·10-s − 13.7·11-s + 56.1·12-s − 56.1·13-s + 59.7·14-s − 43.3·15-s − 77.7·16-s − 34.8·17-s − 281.·18-s + 108.·19-s − 23.4·20-s − 165.·21-s + 50.3·22-s + 23·23-s + 93.2·24-s − 106.·25-s + 206.·26-s + 505.·27-s − 89.5·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 1.95·3-s + 0.688·4-s − 0.380·5-s − 2.54·6-s − 0.877·7-s + 0.404·8-s + 2.84·9-s + 0.494·10-s − 0.375·11-s + 1.34·12-s − 1.19·13-s + 1.14·14-s − 0.746·15-s − 1.21·16-s − 0.496·17-s − 3.69·18-s + 1.30·19-s − 0.262·20-s − 1.71·21-s + 0.488·22-s + 0.208·23-s + 0.792·24-s − 0.855·25-s + 1.55·26-s + 3.60·27-s − 0.604·28-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 | \( 1 + 3.67T + 8T^{2} \) |
| 3 | \( 1 - 10.1T + 27T^{2} \) |
| 5 | \( 1 + 4.25T + 125T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 11 | \( 1 + 13.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 234.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 154.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 174.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 510.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 146.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 238.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 489.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 913.T + 9.12e5T^{2} \) |
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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
−9.600235343478929249664332407687, −8.982920236999197490224624151514, −7.926322420392883445612920615131, −7.58976944358891714218037525400, −6.82703314924171893682026516560, −4.79239607313863453164096987737, −3.61494623204130079340238380338, −2.70817017781302367845888921391, −1.65429601297428286825946840237, 0,
1.65429601297428286825946840237, 2.70817017781302367845888921391, 3.61494623204130079340238380338, 4.79239607313863453164096987737, 6.82703314924171893682026516560, 7.58976944358891714218037525400, 7.926322420392883445612920615131, 8.982920236999197490224624151514, 9.600235343478929249664332407687
