| L(s) = 1 | − 2.27·2-s + 3.17·4-s + 3.81·5-s − 7-s − 2.67·8-s − 8.67·10-s + 2.70·11-s − 3.11·13-s + 2.27·14-s − 0.266·16-s − 5.02·17-s − 0.979·19-s + 12.1·20-s − 6.15·22-s − 1.88·23-s + 9.52·25-s + 7.08·26-s − 3.17·28-s + 8.95·29-s + 6.70·31-s + 5.95·32-s + 11.4·34-s − 3.81·35-s − 5.22·37-s + 2.22·38-s − 10.1·40-s − 0.958·41-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.58·4-s + 1.70·5-s − 0.377·7-s − 0.945·8-s − 2.74·10-s + 0.815·11-s − 0.864·13-s + 0.608·14-s − 0.0667·16-s − 1.21·17-s − 0.224·19-s + 2.70·20-s − 1.31·22-s − 0.392·23-s + 1.90·25-s + 1.39·26-s − 0.600·28-s + 1.66·29-s + 1.20·31-s + 1.05·32-s + 1.96·34-s − 0.644·35-s − 0.858·37-s + 0.361·38-s − 1.61·40-s − 0.149·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140164563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140164563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 3.11T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 0.979T + 19T^{2} \) |
| 23 | \( 1 + 1.88T + 23T^{2} \) |
| 29 | \( 1 - 8.95T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 + 0.958T + 41T^{2} \) |
| 43 | \( 1 - 0.988T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 8.12T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 9.26T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 2.49T + 71T^{2} \) |
| 73 | \( 1 - 5.66T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.160805044372329806312833759471, −7.02273804572322300710624632083, −6.49280090828875773752657864558, −6.33133786259938725958664523531, −5.12327396407703445581464313786, −4.45399357678958672398871108837, −3.00825182021111014562649372048, −2.23857783933979554450018666803, −1.71098463579652186254327470845, −0.67783025877588795294335683334,
0.67783025877588795294335683334, 1.71098463579652186254327470845, 2.23857783933979554450018666803, 3.00825182021111014562649372048, 4.45399357678958672398871108837, 5.12327396407703445581464313786, 6.33133786259938725958664523531, 6.49280090828875773752657864558, 7.02273804572322300710624632083, 8.160805044372329806312833759471
