| L(s) = 1 | + 2.44·2-s − 2.33·3-s + 3.98·4-s + 2.72·5-s − 5.71·6-s − 2.17·7-s + 4.85·8-s + 2.45·9-s + 6.66·10-s − 11-s − 9.30·12-s + 13-s − 5.32·14-s − 6.36·15-s + 3.91·16-s − 4.89·17-s + 5.99·18-s − 7.06·19-s + 10.8·20-s + 5.08·21-s − 2.44·22-s + 7.42·23-s − 11.3·24-s + 2.43·25-s + 2.44·26-s + 1.28·27-s − 8.67·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 1.34·3-s + 1.99·4-s + 1.21·5-s − 2.33·6-s − 0.822·7-s + 1.71·8-s + 0.816·9-s + 2.10·10-s − 0.301·11-s − 2.68·12-s + 0.277·13-s − 1.42·14-s − 1.64·15-s + 0.979·16-s − 1.18·17-s + 1.41·18-s − 1.61·19-s + 2.42·20-s + 1.10·21-s − 0.521·22-s + 1.54·23-s − 2.31·24-s + 0.486·25-s + 0.479·26-s + 0.246·27-s − 1.63·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.967631456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.967631456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.33T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 23 | \( 1 - 7.42T + 23T^{2} \) |
| 29 | \( 1 - 0.696T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 5.26T + 53T^{2} \) |
| 59 | \( 1 + 5.23T + 59T^{2} \) |
| 61 | \( 1 - 3.25T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + 6.15T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−12.83866206677036945333567056231, −12.69961353673509023913925205925, −11.11370455666447179990489686394, −10.76224183702730797627194460599, −9.257306404214687450886310275535, −6.77342333833990462813161532685, −6.22065506304172272720446164836, −5.46601606061398916855192094659, −4.35415927438894319294268437058, −2.50178244710796408026038848820,
2.50178244710796408026038848820, 4.35415927438894319294268437058, 5.46601606061398916855192094659, 6.22065506304172272720446164836, 6.77342333833990462813161532685, 9.257306404214687450886310275535, 10.76224183702730797627194460599, 11.11370455666447179990489686394, 12.69961353673509023913925205925, 12.83866206677036945333567056231
