| L(s) = 1 | + (−1.71 − 1.97i)2-s + (1.45 + 0.936i)3-s + (−0.406 + 2.82i)4-s + (−6.24 − 2.85i)5-s + (−0.645 − 4.48i)6-s + (3.18 − 10.8i)7-s + (−2.52 + 1.62i)8-s + (1.24 + 2.72i)9-s + (5.06 + 17.2i)10-s + (−7.10 − 6.15i)11-s + (−3.23 + 3.73i)12-s + (−11.7 + 3.45i)13-s + (−26.8 + 12.2i)14-s + (−6.42 − 9.99i)15-s + (18.4 + 5.42i)16-s + (31.6 − 4.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.857 − 0.989i)2-s + (0.485 + 0.312i)3-s + (−0.101 + 0.706i)4-s + (−1.24 − 0.570i)5-s + (−0.107 − 0.748i)6-s + (0.454 − 1.54i)7-s + (−0.315 + 0.202i)8-s + (0.138 + 0.303i)9-s + (0.506 + 1.72i)10-s + (−0.645 − 0.559i)11-s + (−0.269 + 0.311i)12-s + (−0.904 + 0.265i)13-s + (−1.92 + 0.877i)14-s + (−0.428 − 0.666i)15-s + (1.15 + 0.339i)16-s + (1.86 − 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.151181 - 0.635154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151181 - 0.635154i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.45 - 0.936i)T \) |
| 23 | \( 1 + (-19.1 + 12.7i)T \) |
| good | 2 | \( 1 + (1.71 + 1.97i)T + (-0.569 + 3.95i)T^{2} \) |
| 5 | \( 1 + (6.24 + 2.85i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (-3.18 + 10.8i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (7.10 + 6.15i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (11.7 - 3.45i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (-31.6 + 4.55i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (3.50 + 0.504i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (0.469 + 3.26i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-23.0 + 14.8i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (-9.63 + 4.40i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (10.0 - 22.0i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (29.7 - 46.2i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 - 43.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-0.0168 + 0.0575i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (19.5 - 5.73i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (54.5 + 84.8i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-68.4 + 59.2i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (41.3 + 47.7i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (4.03 - 28.0i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (-5.14 - 17.5i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-39.5 + 18.0i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (40.4 - 62.9i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (41.7 + 19.0i)T + (6.16e3 + 7.11e3i)T^{2} \) |
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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
−14.05461064112414117142183131721, −12.58146483807592318128486067370, −11.51445505324190221476500462496, −10.58817889269404913431807051577, −9.669307898960209289573252629436, −8.185644163082034364675055810704, −7.64268521543546643356911122448, −4.70519432643192776085618985027, −3.28085613023738327141450293609, −0.75063982528511364704197726034,
3.01656962474016641702358765361, 5.47425412435347733887154937822, 7.22453526866223871512184146308, 7.86242083945883479303557478623, 8.751793765366105842849388218093, 10.07962870479111800143300772670, 11.91996800821287385304617547137, 12.43292685214463483426582804302, 14.61205537021104968571083245693, 15.17355613451804727054699802934
