| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.0358 + 0.0413i)5-s + (−0.959 + 0.281i)6-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.0460 − 0.0296i)10-s + (0.123 + 0.856i)11-s + (0.142 + 0.989i)12-s + (−2.80 + 1.80i)13-s + (0.654 − 0.755i)14-s + (0.0227 − 0.0498i)15-s + (0.841 + 0.540i)16-s + (−3.76 + 1.10i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.0160 + 0.0185i)5-s + (−0.391 + 0.115i)6-s + (0.317 + 0.204i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.0145 − 0.00936i)10-s + (0.0371 + 0.258i)11-s + (0.0410 + 0.285i)12-s + (−0.779 + 0.500i)13-s + (0.175 − 0.201i)14-s + (0.00587 − 0.0128i)15-s + (0.210 + 0.135i)16-s + (−0.914 + 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0903723 + 0.107235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0903723 + 0.107235i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.61 - 1.30i)T \) |
| good | 5 | \( 1 + (-0.0358 - 0.0413i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.123 - 0.856i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.80 - 1.80i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.76 - 1.10i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (7.55 + 2.21i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (5.33 - 1.56i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.02 - 4.44i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.84 + 2.12i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.36 - 5.04i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.932 - 2.04i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 + (-0.00345 - 0.00222i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.04 - 5.16i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 6.21i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.171 - 1.19i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.198 + 1.38i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (1.95 + 0.575i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.18 + 3.33i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (1.02 - 1.18i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.751 - 1.64i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (4.29 + 4.96i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.49234192880674803392743431153, −9.420433525347735996901820330927, −8.702292483165590440301743558256, −7.81328603155712473099283192207, −6.77901323290884381346848674101, −5.99187794777478589867225812513, −4.78180592294577129052657017058, −4.12257302901389822705408559221, −2.50613568418738443612315965668, −1.82603467623467586463064443550,
0.06010898179272020935596729315, 2.25659200706095156260393581198, 3.82458765700320440690412241043, 4.50186555720571243993674967324, 5.51271268691220669361312013836, 6.23660828930282644331841494152, 7.26597166650210321225504135382, 8.068646684357433233828581219872, 8.917655506983506707778485539242, 9.690696682094786085924367704576
