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
−10.14221256626073956188238660980, −9.118470554220582751555774251801, −8.246724874499499248936042381325, −7.12805261250684292422326001891, −6.71557264688496308607883569825, −5.86596479750974698026030880669, −4.57027157750404822185056451659, −3.75912207600423488881306011050, −2.04800760031915239936803526329, −0.848518137445117460935572800292,
1.21526133176025161125044211332, 3.21914913953591173175380564818, 3.88552531718504201928227038660, 5.13179354271727274455652036880, 5.98156576384760999079177740182, 6.47765229398994157368990182337, 7.966861712860991546076900754978, 8.829270627058756503762687873981, 9.448180125285726016835782213583, 10.39858797067513795947455296592