| L(s) = 1 | − 2.68·2-s − 1.17·3-s + 5.21·4-s + 0.884·5-s + 3.14·6-s − 0.361·7-s − 8.62·8-s − 1.62·9-s − 2.37·10-s − 0.487·11-s − 6.11·12-s + 3.81·13-s + 0.969·14-s − 1.03·15-s + 12.7·16-s + 1.24·17-s + 4.36·18-s + 2.14·19-s + 4.60·20-s + 0.423·21-s + 1.31·22-s − 8.74·23-s + 10.1·24-s − 4.21·25-s − 10.2·26-s + 5.42·27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.677·3-s + 2.60·4-s + 0.395·5-s + 1.28·6-s − 0.136·7-s − 3.04·8-s − 0.541·9-s − 0.751·10-s − 0.147·11-s − 1.76·12-s + 1.05·13-s + 0.259·14-s − 0.267·15-s + 3.18·16-s + 0.302·17-s + 1.02·18-s + 0.491·19-s + 1.03·20-s + 0.0924·21-s + 0.279·22-s − 1.82·23-s + 2.06·24-s − 0.843·25-s − 2.00·26-s + 1.04·27-s − 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5072948921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5072948921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 5 | \( 1 - 0.884T + 5T^{2} \) |
| 7 | \( 1 + 0.361T + 7T^{2} \) |
| 11 | \( 1 + 0.487T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 - 7.47T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 7.41T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.990981398321085286966024347805, −7.42242616419701997078446820650, −6.39331572090090696040210704947, −5.99697606243432724840468367133, −5.65366690315908641535024723526, −4.20816269186055289870966502644, −3.11497430766575409083590751283, −2.31609114100914122855430944087, −1.40917950702267257363619722457, −0.51394747001940626558465646389,
0.51394747001940626558465646389, 1.40917950702267257363619722457, 2.31609114100914122855430944087, 3.11497430766575409083590751283, 4.20816269186055289870966502644, 5.65366690315908641535024723526, 5.99697606243432724840468367133, 6.39331572090090696040210704947, 7.42242616419701997078446820650, 7.990981398321085286966024347805
