| L(s) = 1 | + (0.841 − 0.540i)2-s + (1.72 − 0.507i)3-s + (0.415 − 0.909i)4-s + (0.654 + 0.755i)5-s + (1.17 − 1.36i)6-s + (2.15 − 1.38i)7-s + (−0.142 − 0.989i)8-s + (0.202 − 0.130i)9-s + (0.959 + 0.281i)10-s + (−2.28 − 2.63i)11-s + (0.256 − 1.78i)12-s + (0.0540 − 0.376i)13-s + (1.06 − 2.33i)14-s + (1.51 + 0.973i)15-s + (−0.654 − 0.755i)16-s + (−0.252 − 0.553i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (0.997 − 0.292i)3-s + (0.207 − 0.454i)4-s + (0.292 + 0.337i)5-s + (0.481 − 0.555i)6-s + (0.815 − 0.524i)7-s + (−0.0503 − 0.349i)8-s + (0.0674 − 0.0433i)9-s + (0.303 + 0.0890i)10-s + (−0.688 − 0.794i)11-s + (0.0739 − 0.514i)12-s + (0.0150 − 0.104i)13-s + (0.284 − 0.623i)14-s + (0.391 + 0.251i)15-s + (−0.163 − 0.188i)16-s + (−0.0613 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.65631 - 1.48093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65631 - 1.48093i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (7.95 - 1.90i)T \) |
| good | 3 | \( 1 + (-1.72 + 0.507i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (-2.15 + 1.38i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (2.28 + 2.63i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0540 + 0.376i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (0.252 + 0.553i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.11 - 1.99i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (1.45 - 0.428i)T + (19.3 - 12.4i)T^{2} \) |
| 29 | \( 1 - 6.21T + 29T^{2} \) |
| 31 | \( 1 + (-0.791 - 5.50i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + (0.581 + 1.27i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.49 - 7.66i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (8.08 - 2.37i)T + (39.5 - 25.4i)T^{2} \) |
| 53 | \( 1 + (-5.37 + 11.7i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.13 + 7.92i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (4.24 - 4.90i)T + (-8.68 - 60.3i)T^{2} \) |
| 71 | \( 1 + (1.52 - 3.33i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (0.799 - 0.923i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.0399 + 0.277i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.26 - 6.07i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.20 - 0.353i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + 9.82T + 97T^{2} \) |
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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
−10.52407351641171125655076676563, −9.613275634366307508313933802741, −8.421114530052504509972119786603, −7.915172067979315939287588381846, −6.89557219975653160082683111767, −5.69614258526632201738198552218, −4.79124777771591128573045027428, −3.44465618649470168456031672383, −2.72182645145628617435659119313, −1.47706428925074853045130306918,
2.04616754853111639234622532157, 2.95097410614462794255426419666, 4.28067923607276222281378659174, 5.05912124372105867560411243120, 5.99259330468626571455806356230, 7.30120185804542917382243274238, 8.099443089928681304733888371555, 8.769692073740288235292920634040, 9.583605491020423407004234792598, 10.55594129012411618050780781634
