| L(s) = 1 | + (−1.11 + 1.32i)3-s + (−0.229 − 0.0835i)5-s + (−0.342 + 1.94i)7-s + (−0.532 − 2.95i)9-s + (3.98 − 1.45i)11-s + (3.83 − 3.21i)13-s + (0.365 − 0.212i)15-s + (1.08 + 1.87i)17-s + (−0.276 + 0.478i)19-s + (−2.19 − 2.61i)21-s + (1.16 + 6.58i)23-s + (−3.78 − 3.17i)25-s + (4.51 + 2.57i)27-s + (2.79 + 2.34i)29-s + (0.642 + 3.64i)31-s + ⋯ |
| L(s) = 1 | + (−0.641 + 0.767i)3-s + (−0.102 − 0.0373i)5-s + (−0.129 + 0.733i)7-s + (−0.177 − 0.984i)9-s + (1.20 − 0.437i)11-s + (1.06 − 0.892i)13-s + (0.0944 − 0.0547i)15-s + (0.262 + 0.454i)17-s + (−0.0633 + 0.109i)19-s + (−0.479 − 0.569i)21-s + (0.242 + 1.37i)23-s + (−0.756 − 0.635i)25-s + (0.868 + 0.495i)27-s + (0.519 + 0.435i)29-s + (0.115 + 0.654i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07869 + 0.713834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07869 + 0.713834i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.11 - 1.32i)T \) |
| good | 5 | \( 1 + (0.229 + 0.0835i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.342 - 1.94i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.98 + 1.45i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.83 + 3.21i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.08 - 1.87i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.276 - 0.478i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.16 - 6.58i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.79 - 2.34i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.642 - 3.64i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.461 - 0.800i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.17 - 5.18i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.31 + 0.843i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.26 - 7.17i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.04T + 53T^{2} \) |
| 59 | \( 1 + (9.50 + 3.46i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.896 - 5.08i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.53 + 3.80i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.813 + 1.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.46 + 12.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.71 - 5.63i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.08 - 4.26i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (3.82 - 6.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-18.2 + 6.63i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.39858797067513795947455296592, −9.448180125285726016835782213583, −8.829270627058756503762687873981, −7.966861712860991546076900754978, −6.47765229398994157368990182337, −5.98156576384760999079177740182, −5.13179354271727274455652036880, −3.88552531718504201928227038660, −3.21914913953591173175380564818, −1.21526133176025161125044211332,
0.848518137445117460935572800292, 2.04800760031915239936803526329, 3.75912207600423488881306011050, 4.57027157750404822185056451659, 5.86596479750974698026030880669, 6.71557264688496308607883569825, 7.12805261250684292422326001891, 8.246724874499499248936042381325, 9.118470554220582751555774251801, 10.14221256626073956188238660980
