| L(s) = 1 | + 2.24·2-s − 3-s + 3.04·4-s − 4.30·5-s − 2.24·6-s + 7-s + 2.35·8-s + 9-s − 9.67·10-s − 2.14·11-s − 3.04·12-s + 1.66·13-s + 2.24·14-s + 4.30·15-s − 0.798·16-s + 6.77·17-s + 2.24·18-s + 1.36·19-s − 13.1·20-s − 21-s − 4.81·22-s + 0.0543·23-s − 2.35·24-s + 13.5·25-s + 3.73·26-s − 27-s + 3.04·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.52·4-s − 1.92·5-s − 0.917·6-s + 0.377·7-s + 0.834·8-s + 0.333·9-s − 3.06·10-s − 0.645·11-s − 0.880·12-s + 0.460·13-s + 0.600·14-s + 1.11·15-s − 0.199·16-s + 1.64·17-s + 0.529·18-s + 0.312·19-s − 2.93·20-s − 0.218·21-s − 1.02·22-s + 0.0113·23-s − 0.481·24-s + 2.70·25-s + 0.732·26-s − 0.192·27-s + 0.576·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 - 0.0543T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 7.37T + 43T^{2} \) |
| 47 | \( 1 + 0.881T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 - 3.98T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.50310628632488487343639007213, −6.68433477362416735761354160560, −5.83919855980252290361874778978, −5.15212182514354650773002128947, −4.76057141740291330817468048801, −3.78221030513731431394550647160, −3.59639625307683149441824755056, −2.72296538728469121790295281471, −1.27462558638094133557841747556, 0,
1.27462558638094133557841747556, 2.72296538728469121790295281471, 3.59639625307683149441824755056, 3.78221030513731431394550647160, 4.76057141740291330817468048801, 5.15212182514354650773002128947, 5.83919855980252290361874778978, 6.68433477362416735761354160560, 7.50310628632488487343639007213
