| L(s) = 1 | + (−1.18 + 0.771i)2-s + (0.808 − 1.82i)4-s + (0.197 − 0.542i)5-s + (−1.70 − 0.301i)7-s + (0.454 + 2.79i)8-s + (0.184 + 0.795i)10-s + (1.47 − 0.538i)11-s + (3.98 − 3.34i)13-s + (2.25 − 0.961i)14-s + (−2.69 − 2.95i)16-s + (2.89 − 1.67i)17-s + (−4.73 − 2.73i)19-s + (−0.832 − 0.799i)20-s + (−1.33 + 1.78i)22-s + (−0.802 − 4.55i)23-s + ⋯ |
| L(s) = 1 | + (−0.837 + 0.545i)2-s + (0.404 − 0.914i)4-s + (0.0882 − 0.242i)5-s + (−0.645 − 0.113i)7-s + (0.160 + 0.987i)8-s + (0.0584 + 0.251i)10-s + (0.446 − 0.162i)11-s + (1.10 − 0.927i)13-s + (0.602 − 0.256i)14-s + (−0.673 − 0.739i)16-s + (0.701 − 0.405i)17-s + (−1.08 − 0.626i)19-s + (−0.186 − 0.178i)20-s + (−0.285 + 0.379i)22-s + (−0.167 − 0.948i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.843469 - 0.181115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.843469 - 0.181115i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.18 - 0.771i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.197 + 0.542i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.70 + 0.301i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.47 + 0.538i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.98 + 3.34i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.89 + 1.67i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.73 + 2.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.802 + 4.55i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.08 + 3.67i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-10.1 + 1.79i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.21 - 2.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.45 + 2.93i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.730 - 2.00i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.74 - 9.89i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 4.88iT - 53T^{2} \) |
| 59 | \( 1 + (9.53 + 3.47i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 6.35i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.34 - 1.60i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.78 + 6.55i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.13 - 3.69i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.22 - 3.84i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.67 - 8.11i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (12.0 + 6.96i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.9 - 5.07i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−11.24985051202518940451379230602, −10.44524478726199605694944222358, −9.608944833636382353784932000446, −8.645523900161887253218646399823, −7.960906183871190255828997591527, −6.58780432851666984163456574269, −6.06742930505159977970066974107, −4.66328796753040041450574936013, −2.91482628485406344525216676501, −0.907243267119000683448581041922,
1.55494782315614059320127502622, 3.13395395671521250801988952778, 4.18094923796410839940223736183, 6.21017770038852619792572622326, 6.82318457016117900127184662613, 8.212731727614573630652593896944, 8.892531459829493028698534173750, 9.937153191940764819955743911233, 10.55772565475396197870467077984, 11.63509808460321688054754428251
