| L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.0508 − 0.353i)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 + (0.148 − 0.324i)10-s + (4.94 − 1.45i)11-s + (−0.959 + 0.281i)12-s + (−1.93 + 4.23i)13-s + (0.142 + 0.989i)14-s + (0.233 + 0.269i)15-s + (0.415 + 0.909i)16-s + (−0.0245 + 0.0157i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (0.0227 − 0.158i)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 + (0.0468 − 0.102i)10-s + (1.49 − 0.437i)11-s + (−0.276 + 0.0813i)12-s + (−0.536 + 1.17i)13-s + (0.0380 + 0.264i)14-s + (0.0603 + 0.0696i)15-s + (0.103 + 0.227i)16-s + (−0.00595 + 0.00382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81793 + 1.31722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81793 + 1.31722i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.16 - 2.37i)T \) |
| good | 5 | \( 1 + (-0.0508 + 0.353i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-4.94 + 1.45i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.93 - 4.23i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.0245 - 0.0157i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.765 - 0.491i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.02 + 3.22i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.780 - 0.900i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.127 + 0.884i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.29 - 9.01i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.13 - 3.61i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 6.35T + 47T^{2} \) |
| 53 | \( 1 + (-1.31 - 2.86i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (3.44 - 7.55i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.169 - 0.195i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (12.8 + 3.75i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-6.11 - 1.79i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (9.01 + 5.79i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.01 + 10.9i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.239 + 1.66i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-9.46 + 10.9i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.785 + 5.46i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.19478419674153861183224665944, −9.287334282501491610768751526042, −8.685032458848659852916984592363, −7.46799689815511936209834198123, −6.47253272316114005758664536056, −5.96866210623897070921082598914, −4.74880719929694257262857457736, −4.20991913223355649596221272954, −3.07089579777373864776237989180, −1.54764872692061810100667319632,
0.984389796626785820549982722495, 2.33307984104204128346261497088, 3.56848889804140192776184802890, 4.57275812171377195133805924906, 5.45396055789490046371327094984, 6.50119471956559293651493275829, 7.02221811011266382778918162265, 8.001365377632379071534455637793, 9.058984594765464166887499989143, 10.22239191867631421005814522687
