| L(s) = 1 | + (−0.173 − 0.984i)2-s + (1.31 + 1.12i)3-s + (−0.939 + 0.342i)4-s + (1.38 + 1.16i)5-s + (0.875 − 1.49i)6-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (0.484 + 2.96i)9-s + (0.902 − 1.56i)10-s + (−1.84 + 1.54i)11-s + (−1.62 − 0.602i)12-s + (−0.261 + 1.48i)13-s + (0.173 − 0.984i)14-s + (0.523 + 3.08i)15-s + (0.766 − 0.642i)16-s + (−0.0754 + 0.130i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.762 + 0.647i)3-s + (−0.469 + 0.171i)4-s + (0.618 + 0.518i)5-s + (0.357 − 0.610i)6-s + (0.355 + 0.129i)7-s + (0.176 + 0.306i)8-s + (0.161 + 0.986i)9-s + (0.285 − 0.494i)10-s + (−0.555 + 0.466i)11-s + (−0.468 − 0.173i)12-s + (−0.0724 + 0.410i)13-s + (0.0464 − 0.263i)14-s + (0.135 + 0.795i)15-s + (0.191 − 0.160i)16-s + (−0.0182 + 0.0316i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66463 + 0.327747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66463 + 0.327747i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.31 - 1.12i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 5 | \( 1 + (-1.38 - 1.16i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (1.84 - 1.54i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.261 - 1.48i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.0754 - 0.130i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.10 + 3.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.86 + 1.40i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 8.27i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.97 + 1.81i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.21 + 7.29i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.554 + 3.14i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.56 + 6.35i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (8.07 + 2.93i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 + (0.0725 + 0.0608i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.86 + 1.04i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.629 + 3.57i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.34 + 5.79i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.372 - 0.645i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.10 + 6.26i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.18 - 6.70i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.29 - 5.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.98 - 5.01i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.93664633554979173821558489516, −10.64166039398561530119498427946, −9.595568589231266394930098367539, −8.964048885029401225580523445755, −7.959902810425231246224700276258, −6.80358454727235564060380816302, −5.21655266117063468492976920786, −4.32581258237976148177219945468, −2.90909468404719784181413250380, −2.08952193905802851455956355434,
1.27200171708241654178900691620, 2.89173283471230720671421656881, 4.48740722226802234717598763068, 5.71748900624881962029663333234, 6.55318292477688337241861887749, 7.921207265267838279755366152432, 8.163675061521789150957887107890, 9.319983188372917131821710402624, 10.01438569759229272129771193253, 11.33839193425172889586094217590
