| L(s) = 1 | − 2.63·2-s + 3-s + 4.96·4-s − 1.13·5-s − 2.63·6-s + 1.99·7-s − 7.82·8-s + 9-s + 3.00·10-s + 1.82·11-s + 4.96·12-s + 7.15·13-s − 5.27·14-s − 1.13·15-s + 10.7·16-s + 7.66·17-s − 2.63·18-s + 7.80·19-s − 5.65·20-s + 1.99·21-s − 4.82·22-s + 23-s − 7.82·24-s − 3.70·25-s − 18.8·26-s + 27-s + 9.91·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.48·4-s − 0.508·5-s − 1.07·6-s + 0.754·7-s − 2.76·8-s + 0.333·9-s + 0.949·10-s + 0.550·11-s + 1.43·12-s + 1.98·13-s − 1.40·14-s − 0.293·15-s + 2.68·16-s + 1.85·17-s − 0.622·18-s + 1.79·19-s − 1.26·20-s + 0.435·21-s − 1.02·22-s + 0.208·23-s − 1.59·24-s − 0.741·25-s − 3.70·26-s + 0.192·27-s + 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.203754939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203754939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 + 1.13T + 5T^{2} \) |
| 7 | \( 1 - 1.99T + 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 7.15T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 7.80T + 19T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 0.821T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 + 6.35T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 - 7.52T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.44T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.149789110350991162452698493708, −8.327353952892029856077835981188, −7.80926869620479064146620411521, −7.41866134974135268999707825606, −6.30888994099555043735625488866, −5.44744470769248565294820805910, −3.73939811480661128128131876799, −3.14870523366853599759867533626, −1.56299332592609180941386181846, −1.10745230094542969779487931233,
1.10745230094542969779487931233, 1.56299332592609180941386181846, 3.14870523366853599759867533626, 3.73939811480661128128131876799, 5.44744470769248565294820805910, 6.30888994099555043735625488866, 7.41866134974135268999707825606, 7.80926869620479064146620411521, 8.327353952892029856077835981188, 9.149789110350991162452698493708
