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
−10.22239191867631421005814522687, −9.058984594765464166887499989143, −8.001365377632379071534455637793, −7.02221811011266382778918162265, −6.50119471956559293651493275829, −5.45396055789490046371327094984, −4.57275812171377195133805924906, −3.56848889804140192776184802890, −2.33307984104204128346261497088, −0.984389796626785820549982722495,
1.54764872692061810100667319632, 3.07089579777373864776237989180, 4.20991913223355649596221272954, 4.74880719929694257262857457736, 5.96866210623897070921082598914, 6.47253272316114005758664536056, 7.46799689815511936209834198123, 8.685032458848659852916984592363, 9.287334282501491610768751526042, 10.19478419674153861183224665944