| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (−117. + 68.0i)5-s + (93.5 − 161. i)7-s + 181. i·8-s − 769.·10-s + (1.59e3 + 920. i)11-s + (290. + 503. i)13-s + (916. − 529. i)14-s + (−512. + 886. i)16-s + 1.75e3i·17-s − 7.71e3·19-s + (−3.77e3 − 2.17e3i)20-s + (5.20e3 + 9.01e3i)22-s + (−1.27e4 + 7.35e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.942 + 0.544i)5-s + (0.272 − 0.472i)7-s + 0.353i·8-s − 0.769·10-s + (1.19 + 0.691i)11-s + (0.132 + 0.229i)13-s + (0.333 − 0.192i)14-s + (−0.125 + 0.216i)16-s + 0.357i·17-s − 1.12·19-s + (−0.471 − 0.272i)20-s + (0.489 + 0.847i)22-s + (−1.04 + 0.604i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.033269075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033269075\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (117. - 68.0i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-93.5 + 161. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.59e3 - 920. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-290. - 503. i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 1.75e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.71e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.27e4 - 7.35e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.76e4 + 1.59e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.70e4 + 2.94e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 7.70e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-9.83e3 + 5.67e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-2.51e4 + 4.35e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (2.24e4 + 1.29e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.95e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-3.31e5 + 1.91e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.23e5 - 3.86e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.32e5 + 4.02e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.48e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 5.45e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (5.51e4 - 9.54e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.10e4 - 2.36e4i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.78e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-4.48e5 + 7.76e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−12.14449289651270412302372670875, −11.50573751697778521456614721901, −10.50400506243462700827133738255, −9.074227399913178081751059965133, −7.76627311216416390596653665743, −7.07195485461263586854685928794, −5.95654085461311645220749360913, −4.20365254377836082406134738712, −3.79794367203635946775524932553, −1.88166588210095038349436286948,
0.23304344721031213084842847812, 1.74380972623232787267677881507, 3.46917549839837624646708447334, 4.36703639959571861023820529302, 5.62030442685593088673105250455, 6.83042265828379852335026468494, 8.303499294084954796240758472273, 9.010247021687123412912706362051, 10.53275692559766929017124840589, 11.55259392632215622121593750349
