| L(s) = 1 | + (0.540 + 0.841i)2-s + (−0.554 + 1.64i)3-s + (−0.415 + 0.909i)4-s + (3.21 − 0.943i)5-s + (−1.68 + 0.420i)6-s + (−1.80 + 1.56i)7-s + (−0.989 + 0.142i)8-s + (−2.38 − 1.82i)9-s + (2.53 + 2.19i)10-s + (−0.146 − 0.0943i)11-s + (−1.26 − 1.18i)12-s + (0.435 − 0.502i)13-s + (−2.29 − 0.673i)14-s + (−0.234 + 5.79i)15-s + (−0.654 − 0.755i)16-s + (−1.46 − 3.19i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (−0.320 + 0.947i)3-s + (−0.207 + 0.454i)4-s + (1.43 − 0.421i)5-s + (−0.685 + 0.171i)6-s + (−0.683 + 0.591i)7-s + (−0.349 + 0.0503i)8-s + (−0.794 − 0.606i)9-s + (0.800 + 0.693i)10-s + (−0.0442 − 0.0284i)11-s + (−0.364 − 0.342i)12-s + (0.120 − 0.139i)13-s + (−0.613 − 0.180i)14-s + (−0.0605 + 1.49i)15-s + (−0.163 − 0.188i)16-s + (−0.354 − 0.775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0186 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0186 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.916450 + 0.933678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.916450 + 0.933678i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 3 | \( 1 + (0.554 - 1.64i)T \) |
| 23 | \( 1 + (2.56 + 4.05i)T \) |
| good | 5 | \( 1 + (-3.21 + 0.943i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.80 - 1.56i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (0.146 + 0.0943i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.435 + 0.502i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.46 + 3.19i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-6.08 - 2.78i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-5.54 + 2.53i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.22 + 8.55i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.59 - 8.84i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.791 - 2.69i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (5.73 + 0.824i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 7.88iT - 47T^{2} \) |
| 53 | \( 1 + (7.73 + 8.93i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (7.68 + 6.65i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.619 - 0.0891i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.04 - 6.29i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.12 - 3.31i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.52 - 5.53i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.93 + 3.40i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (11.4 + 3.34i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.16 - 15.0i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.10 + 3.76i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−13.64106216249850439019427745824, −12.58893476051304796805376351252, −11.52827704765969697857080032962, −9.836287071316150806642091747150, −9.627893064980467033908768298602, −8.378493531188854203827080875641, −6.40404508885840583127973566543, −5.72601196994695770793704008675, −4.73607846122297777939317403225, −2.93780159743738721194894173870,
1.64846781284882649592142926345, 3.13599377867022002514007650734, 5.29992182212927661203222489594, 6.29306254798689873945380312655, 7.17976733600893172303242234384, 8.998528365733563966409253053005, 10.12104374758270647901659908391, 10.86194842311169340072985272310, 12.12019516903071582435317350020, 13.04769893588552344779722516249
