| L(s) = 1 | + (−0.173 + 5.19i)3-s + (6.54 + 5.48i)5-s + (11.9 + 4.35i)7-s + (−26.9 − 1.79i)9-s + (−7.56 + 6.34i)11-s + (−6.10 + 34.6i)13-s + (−29.6 + 33.0i)15-s + (−44.7 + 77.4i)17-s + (−1.00 − 1.73i)19-s + (−24.6 + 61.3i)21-s + (33.9 − 12.3i)23-s + (−9.04 − 51.3i)25-s + (13.9 − 139. i)27-s + (40.5 + 230. i)29-s + (113. − 41.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.0333 + 0.999i)3-s + (0.585 + 0.490i)5-s + (0.646 + 0.235i)7-s + (−0.997 − 0.0665i)9-s + (−0.207 + 0.173i)11-s + (−0.130 + 0.738i)13-s + (−0.510 + 0.568i)15-s + (−0.637 + 1.10i)17-s + (−0.0121 − 0.0210i)19-s + (−0.256 + 0.637i)21-s + (0.308 − 0.112i)23-s + (−0.0723 − 0.410i)25-s + (0.0997 − 0.995i)27-s + (0.259 + 1.47i)29-s + (0.655 − 0.238i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.947311 + 1.29453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.947311 + 1.29453i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.173 - 5.19i)T \) |
| good | 5 | \( 1 + (-6.54 - 5.48i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-11.9 - 4.35i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (7.56 - 6.34i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (6.10 - 34.6i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (44.7 - 77.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (1.00 + 1.73i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-33.9 + 12.3i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-40.5 - 230. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-113. + 41.1i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-98.7 + 170. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (4.40 - 25.0i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-410. + 344. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (17.7 + 6.44i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (122. + 102. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-385. - 140. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-152. + 864. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-69.1 + 119. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-554. - 959. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (48.8 + 276. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-168. - 958. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-530. - 919. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (158. - 133. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−13.85896539862644825891336335960, −12.38018906266001909693811429359, −11.11519294451990104049727716743, −10.46416602460810021897548951910, −9.324996618067552082459338168037, −8.339442173347833587384591935602, −6.62368634540876821592443675997, −5.35709708418722174168277214401, −4.10471673610282119490895178543, −2.32405397787771880520850342995,
0.932001144327614861707209639759, 2.56583933066714859361748425337, 4.89428644910370636861231149162, 6.05018545825276246686836958172, 7.42119814339140148151682273903, 8.355214577122052569159710749689, 9.573876780322873035417656794225, 11.02575759235794188488228361061, 11.93037386494232846337520746162, 13.14715391614837655410962915481
