| L(s) = 1 | + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.288 + 2.00i)5-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.841 + 1.84i)10-s + (4.57 + 1.34i)11-s + (0.959 + 0.281i)12-s + (−2.30 − 5.05i)13-s + (−0.142 + 0.989i)14-s + (−1.32 + 1.53i)15-s + (0.415 − 0.909i)16-s + (3.69 + 2.37i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (0.128 + 0.896i)5-s + (0.343 + 0.220i)6-s + (−0.157 + 0.343i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.266 + 0.582i)10-s + (1.37 + 0.404i)11-s + (0.276 + 0.0813i)12-s + (−0.640 − 1.40i)13-s + (−0.0380 + 0.264i)14-s + (−0.342 + 0.395i)15-s + (0.103 − 0.227i)16-s + (0.895 + 0.575i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.57378 + 1.19837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.57378 + 1.19837i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-0.181 - 4.79i)T \) |
| good | 5 | \( 1 + (-0.288 - 2.00i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-4.57 - 1.34i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.30 + 5.05i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.69 - 2.37i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.65 - 1.70i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (1.20 + 0.773i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.563 - 0.649i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.70 - 11.8i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.966 + 6.72i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.54 - 2.93i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 + (-0.00208 + 0.00457i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.96 + 6.50i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-7.15 + 8.25i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (1.31 - 0.384i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-9.03 + 2.65i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 0.958i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.84 + 8.42i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.43 + 16.9i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (11.7 + 13.6i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.60 + 11.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.09146081106135070840918769474, −9.664704921112138548070764574817, −8.448227446534999030849569162310, −7.52213783611010625232025716419, −6.58208368284553785461518341154, −5.79159580373628342977165262021, −4.79956086571663480949071302819, −3.57411636703529739283190172975, −3.08626652828268990775326835486, −1.77609269828685795397764839679,
1.13382904535278310245565893081, 2.40313421877139561235337620680, 3.80953683102409940956916376679, 4.47152435372829515064194367665, 5.53601844234136768455771133702, 6.66589644932884192827636432228, 7.06356974231925794626585848830, 8.257094080778835833061857950875, 9.074549843100328427973214881341, 9.569992627798817154589756186530
