| L(s) = 1 | + (1.44 − 1.72i)2-s + (−0.130 + 1.72i)3-s + (−0.529 − 3.00i)4-s + (−0.536 − 2.17i)5-s + (2.78 + 2.71i)6-s + (3.05 + 0.538i)7-s + (−2.03 − 1.17i)8-s + (−2.96 − 0.451i)9-s + (−4.51 − 2.21i)10-s + (−3.82 + 1.39i)11-s + (5.25 − 0.522i)12-s + (0.913 + 1.08i)13-s + (5.33 − 4.47i)14-s + (3.81 − 0.643i)15-s + (0.757 − 0.275i)16-s + (−5.52 + 3.18i)17-s + ⋯ |
| L(s) = 1 | + (1.02 − 1.21i)2-s + (−0.0754 + 0.997i)3-s + (−0.264 − 1.50i)4-s + (−0.239 − 0.970i)5-s + (1.13 + 1.11i)6-s + (1.15 + 0.203i)7-s + (−0.720 − 0.416i)8-s + (−0.988 − 0.150i)9-s + (−1.42 − 0.699i)10-s + (−1.15 + 0.419i)11-s + (1.51 − 0.150i)12-s + (0.253 + 0.301i)13-s + (1.42 − 1.19i)14-s + (0.986 − 0.166i)15-s + (0.189 − 0.0689i)16-s + (−1.33 + 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43982 - 0.872333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43982 - 0.872333i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.130 - 1.72i)T \) |
| 5 | \( 1 + (0.536 + 2.17i)T \) |
| good | 2 | \( 1 + (-1.44 + 1.72i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-3.05 - 0.538i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.82 - 1.39i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.913 - 1.08i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.52 - 3.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.59 - 2.76i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.75 + 0.838i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.87 + 2.41i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.453 - 2.57i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.545 + 0.314i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.50 + 3.78i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.01 + 8.29i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-8.23 - 1.45i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 0.0211iT - 53T^{2} \) |
| 59 | \( 1 + (-1.27 - 0.463i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.492 + 2.79i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.983 + 1.17i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.47 + 12.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.31 - 3.64i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.125 + 0.105i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.27 - 2.70i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.60 - 13.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.59 - 12.6i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.89680450560128709350739718191, −12.01296186215801993290666910645, −11.04340222247660370244385641802, −10.52154718749337591489451944465, −9.089275436411586909050009957484, −8.108181532170416224551869441378, −5.53677231621917725598275305919, −4.76124924528586337418719614142, −3.99722988898251141874106648008, −2.13581053554971446029692622815,
2.78365435536730563659046819413, 4.68461924641340682415152918209, 5.81052576421503902365652247490, 6.96323234264290569070025913669, 7.58618113103374676835183908528, 8.491491294292865719438190404832, 10.89193703654574570314114576758, 11.40943669281785388088070357772, 13.00709625245510140269866951290, 13.48234359155179730866626400106
