L(s) = 1 | + (−1.04 − 1.24i)2-s + (−0.107 + 0.611i)4-s + (2.12 − 0.705i)5-s + (1.93 + 1.11i)7-s + (−1.93 + 1.11i)8-s + (−3.08 − 1.89i)10-s + (2.82 + 4.88i)11-s + (−1.60 − 4.41i)13-s + (−0.627 − 3.55i)14-s + (4.56 + 1.66i)16-s + (−0.505 − 0.601i)17-s + (2.63 + 3.47i)19-s + (0.202 + 1.37i)20-s + (3.12 − 8.58i)22-s + (5.05 + 0.890i)23-s + ⋯ |
L(s) = 1 | + (−0.735 − 0.877i)2-s + (−0.0539 + 0.305i)4-s + (0.948 − 0.315i)5-s + (0.730 + 0.421i)7-s + (−0.683 + 0.394i)8-s + (−0.975 − 0.600i)10-s + (0.850 + 1.47i)11-s + (−0.445 − 1.22i)13-s + (−0.167 − 0.951i)14-s + (1.14 + 0.415i)16-s + (−0.122 − 0.146i)17-s + (0.603 + 0.797i)19-s + (0.0453 + 0.307i)20-s + (0.666 − 1.83i)22-s + (1.05 + 0.185i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23610 - 0.648441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23610 - 0.648441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.705i)T \) |
| 19 | \( 1 + (-2.63 - 3.47i)T \) |
good | 2 | \( 1 + (1.04 + 1.24i)T + (-0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.93 - 1.11i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.82 - 4.88i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.60 + 4.41i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.505 + 0.601i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-5.05 - 0.890i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.32 - 1.95i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.05 - 7.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.985iT - 37T^{2} \) |
| 41 | \( 1 + (-1.33 - 0.484i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.49 + 0.264i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.617 - 0.735i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (4.34 + 0.766i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.56 + 6.34i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 2.06i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.08 - 2.48i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.24 + 7.06i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.18 + 14.2i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.25 - 0.458i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (6.51 + 3.76i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.19 - 1.16i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.20 - 9.77i)T + (-16.8 + 95.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02282069210267817512251803912, −9.376779878349954230579020094676, −8.744211176548132847808927732142, −7.73910408178956541241874160278, −6.62087171023099359395898334718, −5.42459968556465252230927826896, −4.90729166718299853327113675138, −3.15995374652500352998754843882, −2.01544517849142661390654092972, −1.27345145443751765146952403585,
1.08849147185354809027184760750, 2.69074318300692935693934996754, 4.00538639983583070795255960180, 5.33078044257250343479952688049, 6.27527085311544188985603171202, 6.88360191324502469738289914675, 7.67960288982385084893792953874, 8.841698344164781018453460162635, 9.102839067680749867743416029558, 10.00454634673303690121830842217