| 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
−9.569992627798817154589756186530, −9.074549843100328427973214881341, −8.257094080778835833061857950875, −7.06356974231925794626585848830, −6.66589644932884192827636432228, −5.53601844234136768455771133702, −4.47152435372829515064194367665, −3.80953683102409940956916376679, −2.40313421877139561235337620680, −1.13382904535278310245565893081,
1.77609269828685795397764839679, 3.08626652828268990775326835486, 3.57411636703529739283190172975, 4.79956086571663480949071302819, 5.79159580373628342977165262021, 6.58208368284553785461518341154, 7.52213783611010625232025716419, 8.448227446534999030849569162310, 9.664704921112138548070764574817, 10.09146081106135070840918769474
