| L(s) = 1 | − 0.837·2-s − 3-s − 1.29·4-s + 2.34·5-s + 0.837·6-s + 7-s + 2.76·8-s + 9-s − 1.96·10-s + 2.64·11-s + 1.29·12-s − 4.81·13-s − 0.837·14-s − 2.34·15-s + 0.282·16-s − 7.16·17-s − 0.837·18-s + 4.98·19-s − 3.04·20-s − 21-s − 2.21·22-s + 3.89·23-s − 2.76·24-s + 0.515·25-s + 4.02·26-s − 27-s − 1.29·28-s + ⋯ |
| L(s) = 1 | − 0.592·2-s − 0.577·3-s − 0.649·4-s + 1.05·5-s + 0.341·6-s + 0.377·7-s + 0.976·8-s + 0.333·9-s − 0.622·10-s + 0.798·11-s + 0.374·12-s − 1.33·13-s − 0.223·14-s − 0.606·15-s + 0.0706·16-s − 1.73·17-s − 0.197·18-s + 1.14·19-s − 0.681·20-s − 0.218·21-s − 0.473·22-s + 0.811·23-s − 0.563·24-s + 0.103·25-s + 0.790·26-s − 0.192·27-s − 0.245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012249818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012249818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.837T + 2T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 7.16T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 + 9.78T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 - 4.10T + 59T^{2} \) |
| 61 | \( 1 + 0.167T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 - 8.35T + 97T^{2} \) |
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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
−7.64587951435694558715370824051, −7.27374600160340092663343756528, −6.53053946643954659824597919857, −5.54677214405831565572864534717, −5.19124737631055498223412064585, −4.45459848059338382921766536433, −3.67257079847768248127957229024, −2.24938097827885358308783271356, −1.69217785635148187797227872860, −0.57375623304922053681235457900,
0.57375623304922053681235457900, 1.69217785635148187797227872860, 2.24938097827885358308783271356, 3.67257079847768248127957229024, 4.45459848059338382921766536433, 5.19124737631055498223412064585, 5.54677214405831565572864534717, 6.53053946643954659824597919857, 7.27374600160340092663343756528, 7.64587951435694558715370824051
