| L(s) = 1 | + (0.766 − 0.642i)2-s + (0.586 + 1.62i)3-s + (0.173 − 0.984i)4-s + (0.483 − 2.74i)5-s + (1.49 + 0.871i)6-s + (0.286 − 2.63i)7-s + (−0.500 − 0.866i)8-s + (−2.31 + 1.91i)9-s + (−1.39 − 2.41i)10-s + (−0.275 − 1.56i)11-s + (1.70 − 0.294i)12-s + (0.943 − 5.34i)13-s + (−1.47 − 2.19i)14-s + (4.75 − 0.820i)15-s + (−0.939 − 0.342i)16-s + (3.17 + 5.49i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.338 + 0.940i)3-s + (0.0868 − 0.492i)4-s + (0.216 − 1.22i)5-s + (0.611 + 0.355i)6-s + (0.108 − 0.994i)7-s + (−0.176 − 0.306i)8-s + (−0.770 + 0.637i)9-s + (−0.440 − 0.762i)10-s + (−0.0830 − 0.471i)11-s + (0.492 − 0.0849i)12-s + (0.261 − 1.48i)13-s + (−0.393 − 0.587i)14-s + (1.22 − 0.211i)15-s + (−0.234 − 0.0855i)16-s + (0.769 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76635 - 0.962057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76635 - 0.962057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.586 - 1.62i)T \) |
| 7 | \( 1 + (-0.286 + 2.63i)T \) |
| good | 5 | \( 1 + (-0.483 + 2.74i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.275 + 1.56i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.943 + 5.34i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.17 - 5.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.31 - 5.74i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.11 - 5.13i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.0620 + 0.351i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.432 - 2.45i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 + (0.0866 - 0.491i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-9.13 + 7.66i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.92 - 10.9i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.02 + 5.24i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.96 - 3.26i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.975 + 5.53i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 1.47i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0982 + 0.170i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.65T + 73T^{2} \) |
| 79 | \( 1 + (-1.98 + 1.66i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.07 - 17.4i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-0.662 + 1.14i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.75 - 5.66i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.78071318684220197487321223718, −10.59365347695426130386310549483, −9.563810605256805949400515076650, −8.519715301765921913110713061986, −7.81348423502136085937937232296, −5.83455774072062358433929772335, −5.22717616509957256162136631523, −4.06160924497335620359755138730, −3.32755906694659154475687007955, −1.27239342133914464589921182703,
2.30372660730254340114351345058, 2.96479562624401785793247995284, 4.72006977744561376235342313197, 6.04312276831697739693885445989, 6.86868327084318732155221753798, 7.28555094989934290273325314734, 8.715401878560489712586572362775, 9.330837854148333846005701731508, 10.94812292106151142702562832082, 11.67838498733348374337137878832
