| L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.376 + 1.69i)3-s + (−0.173 + 0.984i)4-s + (1.90 − 1.17i)5-s + (1.53 − 0.798i)6-s + (0.531 − 0.0936i)7-s + (0.866 − 0.500i)8-s + (−2.71 − 1.27i)9-s + (−2.12 − 0.705i)10-s + (3.82 + 1.39i)11-s + (−1.59 − 0.663i)12-s + (−1.58 + 1.88i)13-s + (−0.413 − 0.346i)14-s + (1.26 + 3.66i)15-s + (−0.939 − 0.342i)16-s + (3.71 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (−0.217 + 0.976i)3-s + (−0.0868 + 0.492i)4-s + (0.851 − 0.524i)5-s + (0.627 − 0.326i)6-s + (0.200 − 0.0353i)7-s + (0.306 − 0.176i)8-s + (−0.905 − 0.423i)9-s + (−0.671 − 0.222i)10-s + (1.15 + 0.419i)11-s + (−0.461 − 0.191i)12-s + (−0.439 + 0.523i)13-s + (−0.110 − 0.0926i)14-s + (0.326 + 0.945i)15-s + (−0.234 − 0.0855i)16-s + (0.901 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10416 + 0.168262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10416 + 0.168262i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 + 0.766i)T \) |
| 3 | \( 1 + (0.376 - 1.69i)T \) |
| 5 | \( 1 + (-1.90 + 1.17i)T \) |
| good | 7 | \( 1 + (-0.531 + 0.0936i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.82 - 1.39i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.58 - 1.88i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.71 - 2.14i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.08 - 1.88i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.39 - 0.775i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.558 - 0.468i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.28 - 7.30i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (6.45 + 3.72i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.82 + 4.88i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.73 + 7.50i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-9.13 + 1.61i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.116iT - 53T^{2} \) |
| 59 | \( 1 + (10.2 - 3.73i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.35 + 7.66i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.14 - 7.32i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.98 + 12.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.27 - 1.31i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.10 - 3.44i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (9.14 + 10.8i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.79 + 8.30i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.646 - 1.77i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−12.00872392142085786381065259793, −10.76372661081448765888047298361, −10.06288716545551395494026053628, −9.202576658336700488878238731607, −8.706549051460442262545547385167, −7.06129816290378990960926894172, −5.69423565620519581493007101628, −4.65450098708092065024131269665, −3.45534653237852983676406822426, −1.62073856234891363423406184014,
1.27785797638270541661792711958, 2.89403335642523723194634574598, 5.18524250809725663640522741506, 6.10641106290122583649813996581, 6.92370669265723806186436957095, 7.78061357876791152742104844397, 8.954932633331397039036349944511, 9.811982027895351786823747415166, 10.98683020796435987857711244495, 11.76934741093854505702633449394
