| L(s) = 1 | + (0.173 + 0.984i)2-s + (−1.73 − 0.0288i)3-s + (−0.939 + 0.342i)4-s + (1.44 + 1.21i)5-s + (−0.272 − 1.71i)6-s + (0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (2.99 + 0.100i)9-s + (−0.941 + 1.63i)10-s + (−2.52 + 2.11i)11-s + (1.63 − 0.565i)12-s + (−0.345 + 1.96i)13-s + (−0.173 + 0.984i)14-s + (−2.46 − 2.13i)15-s + (0.766 − 0.642i)16-s + (−1.72 + 2.98i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.999 − 0.0166i)3-s + (−0.469 + 0.171i)4-s + (0.645 + 0.541i)5-s + (−0.111 − 0.698i)6-s + (0.355 + 0.129i)7-s + (−0.176 − 0.306i)8-s + (0.999 + 0.0333i)9-s + (−0.297 + 0.515i)10-s + (−0.761 + 0.638i)11-s + (0.472 − 0.163i)12-s + (−0.0959 + 0.544i)13-s + (−0.0464 + 0.263i)14-s + (−0.636 − 0.552i)15-s + (0.191 − 0.160i)16-s + (−0.417 + 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.315525 + 0.878010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.315525 + 0.878010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (1.73 + 0.0288i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 5 | \( 1 + (-1.44 - 1.21i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (2.52 - 2.11i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.345 - 1.96i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.72 - 2.98i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 1.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.20 - 1.16i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.895 - 5.07i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (8.22 - 2.99i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 3.61i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.18 - 6.69i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.71 + 3.96i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.423 - 0.154i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + (-6.94 - 5.82i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-12.7 - 4.65i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.997 + 5.65i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.17 + 3.76i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.36 + 12.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.07 + 11.7i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.508 - 2.88i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-0.135 - 0.235i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.24 - 1.04i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.76993355142354154741504851572, −10.62735155153625364875546137135, −10.11256828898090260816621154048, −8.967578709453441303387709727965, −7.64895636697090325168612858958, −6.87817581823423885013883979088, −5.93014518969294105752054122013, −5.19213788268252627856475513402, −4.04621041165899798750905703485, −1.99673902591658931367177098777,
0.68394596072725699033237983099, 2.31675591900469192682570380900, 4.07422077800139305258417017486, 5.28763335589803899371019333857, 5.67010700006293185261973954175, 7.15394862320821816948537171218, 8.359858913779399171180677089833, 9.521894689415002847499407863985, 10.22482704497083704344061907734, 11.15772620763585644985281687428
