| 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
−11.67838498733348374337137878832, −10.94812292106151142702562832082, −9.330837854148333846005701731508, −8.715401878560489712586572362775, −7.28555094989934290273325314734, −6.86868327084318732155221753798, −6.04312276831697739693885445989, −4.72006977744561376235342313197, −2.96479562624401785793247995284, −2.30372660730254340114351345058,
1.27239342133914464589921182703, 3.32755906694659154475687007955, 4.06160924497335620359755138730, 5.22717616509957256162136631523, 5.83455774072062358433929772335, 7.81348423502136085937937232296, 8.519715301765921913110713061986, 9.563810605256805949400515076650, 10.59365347695426130386310549483, 10.78071318684220197487321223718
