| L(s) = 1 | + (−0.633 + 1.61i)2-s + (−0.0747 + 0.997i)3-s + (−0.738 − 0.685i)4-s + (2.02 + 1.38i)5-s + (−1.56 − 0.752i)6-s + (2.55 − 0.669i)7-s + (−1.55 + 0.746i)8-s + (−0.988 − 0.149i)9-s + (−3.51 + 2.39i)10-s + (−1.16 + 0.175i)11-s + (0.738 − 0.685i)12-s + (−3.07 − 3.85i)13-s + (−0.541 + 4.55i)14-s + (−1.52 + 1.91i)15-s + (−0.373 − 4.98i)16-s + (−0.121 − 0.0374i)17-s + ⋯ |
| L(s) = 1 | + (−0.448 + 1.14i)2-s + (−0.0431 + 0.575i)3-s + (−0.369 − 0.342i)4-s + (0.905 + 0.617i)5-s + (−0.637 − 0.307i)6-s + (0.967 − 0.253i)7-s + (−0.548 + 0.263i)8-s + (−0.329 − 0.0496i)9-s + (−1.11 + 0.757i)10-s + (−0.350 + 0.0528i)11-s + (0.213 − 0.197i)12-s + (−0.852 − 1.06i)13-s + (−0.144 + 1.21i)14-s + (−0.394 + 0.494i)15-s + (−0.0934 − 1.24i)16-s + (−0.0294 − 0.00907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.428586 + 0.939267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428586 + 0.939267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0747 - 0.997i)T \) |
| 7 | \( 1 + (-2.55 + 0.669i)T \) |
| good | 2 | \( 1 + (0.633 - 1.61i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.02 - 1.38i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (1.16 - 0.175i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (3.07 + 3.85i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.121 + 0.0374i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.786 - 1.36i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.92 + 1.51i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.371 - 1.62i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.98 + 6.48i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (5.76 - 2.77i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-9.98 - 4.80i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.92 + 7.45i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (8.18 + 7.59i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (5.88 - 4.01i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (9.05 - 8.40i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (2.53 - 4.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.08 + 9.15i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (3.24 + 8.26i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-6.33 - 10.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (10.4 - 13.0i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-5.91 - 0.891i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 1.74T + 97T^{2} \) |
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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
−13.91425607321615859325588458377, −12.47326351501808482897076868282, −11.05475079688242193567300065055, −10.25333145356931387587713711275, −9.198283828584557527624505244934, −8.030333307822750145434968665712, −7.15852678578062314164215038562, −5.85019896149605368204981713723, −4.98174189007976339161513602351, −2.76205891101267559132202009455,
1.46499540180019923755636764041, 2.50542939218154527936843698932, 4.77842106401632364450976647353, 6.05152104860861114598788270865, 7.57364729215316959532944151734, 8.991787926060262187013910503827, 9.516416725525196552498870749816, 10.85745817482932375044319903113, 11.63258301895628985453662030279, 12.48325535635316909143133004242
