| L(s) = 1 | + (−1.67 + 15.4i)3-s − 6.76i·5-s − 132. i·7-s + (−237. − 51.9i)9-s + 687.·11-s − 609.·13-s + (104. + 11.3i)15-s − 550. i·17-s + 232. i·19-s + (2.05e3 + 222. i)21-s − 4.48e3·23-s + 3.07e3·25-s + (1.20e3 − 3.59e3i)27-s + 2.04e3i·29-s + 6.67e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.107 + 0.994i)3-s − 0.121i·5-s − 1.02i·7-s + (−0.976 − 0.213i)9-s + 1.71·11-s − 1.00·13-s + (0.120 + 0.0130i)15-s − 0.462i·17-s + 0.147i·19-s + (1.01 + 0.109i)21-s − 1.76·23-s + 0.985·25-s + (0.317 − 0.948i)27-s + 0.451i·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5256615424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5256615424\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 - 15.4i)T \) |
| good | 5 | \( 1 + 6.76iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 132. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 687.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 609.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 550. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 232. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 4.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.04e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.67e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 9.78e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.24e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.33e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.81e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.49e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.92e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−10.84572088775911055889895571156, −9.961097730937313321937639215162, −9.381398470893954091949918140153, −8.353049354240057950389747585892, −7.12121943102097817645566001732, −6.21464773456231671744240293067, −4.82913309405766179317182395705, −4.15852968540993900379538357278, −3.13241231832374912052013793613, −1.31303116323558031699306147925,
0.13268175328681849270870252967, 1.64708637757671843200771524579, 2.52243206946436051820780285824, 4.00457581195938065421540812856, 5.50037348034997690200800415256, 6.31100616962092341020090458528, 7.12013538904564902120748622205, 8.244376994930270752752022149182, 9.008470701311584773716107759495, 9.964952814716760617164400827315
