| L(s) = 1 | + (−1.04 + 0.953i)2-s + (−1.77 + 1.53i)3-s + (0.183 − 1.99i)4-s + (3.72 − 0.536i)5-s + (0.388 − 3.29i)6-s + (1.80 − 3.95i)7-s + (1.70 + 2.25i)8-s + (0.356 − 2.48i)9-s + (−3.38 + 4.11i)10-s + (−0.791 + 2.69i)11-s + (2.73 + 3.81i)12-s + (1.31 − 0.602i)13-s + (1.88 + 5.85i)14-s + (−5.79 + 6.68i)15-s + (−3.93 − 0.729i)16-s + (0.426 + 0.274i)17-s + ⋯ |
| L(s) = 1 | + (−0.738 + 0.673i)2-s + (−1.02 + 0.887i)3-s + (0.0915 − 0.995i)4-s + (1.66 − 0.239i)5-s + (0.158 − 1.34i)6-s + (0.682 − 1.49i)7-s + (0.603 + 0.797i)8-s + (0.118 − 0.826i)9-s + (−1.07 + 1.30i)10-s + (−0.238 + 0.812i)11-s + (0.789 + 1.10i)12-s + (0.365 − 0.167i)13-s + (0.503 + 1.56i)14-s + (−1.49 + 1.72i)15-s + (−0.983 − 0.182i)16-s + (0.103 + 0.0664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.746186 + 0.363268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.746186 + 0.363268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 - 0.953i)T \) |
| 23 | \( 1 + (-4.04 + 2.57i)T \) |
| good | 3 | \( 1 + (1.77 - 1.53i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-3.72 + 0.536i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 3.95i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.791 - 2.69i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.31 + 0.602i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.426 - 0.274i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 1.69i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.637 - 0.991i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (5.28 - 6.09i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.78 + 0.256i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 7.38i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.318 - 0.275i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + (8.99 + 4.10i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.14 + 0.524i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (6.78 + 5.87i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (0.439 + 1.49i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (3.81 - 1.11i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (3.83 - 2.46i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (4.36 + 9.55i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (9.43 + 1.35i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.96 + 9.18i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.199 - 1.38i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.92720928072881738175959871246, −11.14300803894372054493537560678, −10.41644310671246218552884292004, −10.06463965880027815725117593140, −9.011667694224231919246037707894, −7.49344954898550194766789047317, −6.38582268924028983550117162093, −5.31978355357398811763349579344, −4.63404375197393323642001067602, −1.43826837530141691061907206999,
1.51436339012439243519172205497, 2.61634591858701826273604060032, 5.45828919231135934537946812199, 5.99470447420943853218484210613, 7.28187419701054692600659559335, 8.761559549878600283916739737594, 9.403427142967182804985974682639, 10.80913698032353674323245649981, 11.36956141308599831175636696153, 12.31530298255563444588836499841
