| L(s) = 1 | + (0.446 − 0.324i)2-s + (0.927 + 2.85i)3-s + (−2.37 + 7.31i)4-s + (9.46 + 6.87i)5-s + (1.33 + 0.972i)6-s + (−2.16 + 6.65i)7-s + (2.67 + 8.23i)8-s + (−7.28 + 5.29i)9-s + 6.45·10-s + (35.2 + 9.37i)11-s − 23.0·12-s + (−7.29 + 5.30i)13-s + (1.19 + 3.67i)14-s + (−10.8 + 33.3i)15-s + (−45.9 − 33.3i)16-s + (17.8 + 12.9i)17-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.114i)2-s + (0.178 + 0.549i)3-s + (−0.297 + 0.914i)4-s + (0.846 + 0.615i)5-s + (0.0911 + 0.0661i)6-s + (−0.116 + 0.359i)7-s + (0.118 + 0.363i)8-s + (−0.269 + 0.195i)9-s + 0.204·10-s + (0.966 + 0.256i)11-s − 0.555·12-s + (−0.155 + 0.113i)13-s + (0.0227 + 0.0701i)14-s + (−0.186 + 0.574i)15-s + (−0.717 − 0.521i)16-s + (0.254 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.890719 + 1.81622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.890719 + 1.81622i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.927 - 2.85i)T \) |
| 7 | \( 1 + (2.16 - 6.65i)T \) |
| 11 | \( 1 + (-35.2 - 9.37i)T \) |
| good | 2 | \( 1 + (-0.446 + 0.324i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (-9.46 - 6.87i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (7.29 - 5.30i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-17.8 - 12.9i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (5.29 + 16.3i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 69.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-6.39 + 19.6i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (89.7 - 65.2i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (32.0 - 98.6i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-31.8 - 98.0i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-93.0 - 286. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (57.0 - 41.4i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-194. + 599. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-541. - 393. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 653.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (525. + 381. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-33.4 + 102. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (409. - 297. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (55.2 + 40.1i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 337.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-551. + 400. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.10295650452472430536258151754, −11.19480135110728817496700087254, −9.995297332327203315471834013883, −9.281362397389747340238875522146, −8.322884030807921405979746646266, −7.03668826037845681620216184824, −5.91462201383556693638299027211, −4.52239113021580842668951216785, −3.39941096718401197331260586270, −2.21530231553164432425640994422,
0.793759441732570591006067861387, 1.90983642047373156784121195562, 3.96131934551655119537486319606, 5.36652464527079758852245637880, 6.12332927921054366465847740405, 7.19053959848912613502865416008, 8.667804184526058781595787394077, 9.448028539862748958929752858936, 10.23385503319091800840478831964, 11.45839952140012230898999391910
