| L(s) = 1 | + (2.47 + 7.60i)2-s + (−27.4 + 84.4i)3-s + (−51.7 + 37.6i)4-s − 131.·5-s − 710.·6-s + (1.04e3 − 755. i)7-s + (−414. − 300. i)8-s + (−4.61e3 − 3.34e3i)9-s + (−323. − 996. i)10-s + (−5.25e3 + 3.81e3i)11-s + (−1.75e3 − 5.40e3i)12-s + (2.29e3 − 7.06e3i)13-s + (8.32e3 + 6.04e3i)14-s + (3.59e3 − 1.10e4i)15-s + (1.26e3 − 3.89e3i)16-s + (−620. − 450. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.586 + 1.80i)3-s + (−0.404 + 0.293i)4-s − 0.468·5-s − 1.34·6-s + (1.14 − 0.833i)7-s + (−0.286 − 0.207i)8-s + (−2.10 − 1.53i)9-s + (−0.102 − 0.315i)10-s + (−1.19 + 0.865i)11-s + (−0.293 − 0.902i)12-s + (0.289 − 0.891i)13-s + (0.810 + 0.589i)14-s + (0.275 − 0.846i)15-s + (0.0772 − 0.237i)16-s + (−0.0306 − 0.0222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.140559 - 0.0485489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140559 - 0.0485489i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.47 - 7.60i)T \) |
| 31 | \( 1 + (151. + 1.65e5i)T \) |
| good | 3 | \( 1 + (27.4 - 84.4i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 5 | \( 1 + 131.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-1.04e3 + 755. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (5.25e3 - 3.81e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-2.29e3 + 7.06e3i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (620. + 450. i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (176. + 544. i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-1.62e4 - 1.18e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-7.64e4 - 2.35e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 37 | \( 1 + 5.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (8.32e4 + 2.56e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + (1.23e5 + 3.81e5i)T + (-2.19e11 + 1.59e11i)T^{2} \) |
| 47 | \( 1 + (-3.36e5 + 1.03e6i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (8.59e4 + 6.24e4i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (5.81e5 - 1.78e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + 1.64e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.63e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (3.79e6 + 2.75e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-2.55e6 + 1.85e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-1.29e6 - 9.37e5i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (2.98e5 + 9.18e5i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-5.28e6 + 3.84e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (6.58e6 - 4.78e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.80335445547638970418848839669, −12.09227117536856477451342020193, −10.77832814374954436829143252761, −10.26043233310285643982954250265, −8.662201296745838106342958034928, −7.46041713556595061567820096972, −5.40212183495454280927926918236, −4.73914534827268951499595892786, −3.59488902316748574112551068846, −0.05593992987468880526871700894,
1.43214488694837420071285079049, 2.56483231114099465295885993871, 5.06323793024900225317388339030, 6.19111634594318475041758711590, 7.82865730587326026639694811416, 8.524252558694757108441458781814, 10.94452676204780436607351443551, 11.61726711265397765208187872316, 12.30740034433547007223073644688, 13.46337177441038784772392076973
