| L(s) = 1 | + (7.51 − 2.73i)2-s + (−0.665 + 3.77i)3-s + (49.0 − 41.1i)4-s + (10.1 + 8.52i)5-s + (5.32 + 30.2i)6-s + (−800. − 1.38e3i)7-s + (256. − 443. i)8-s + (2.04e3 + 742. i)9-s + (99.7 + 36.3i)10-s + (706. − 1.22e3i)11-s + (122. + 212. i)12-s + (−1.78e3 − 1.01e4i)13-s + (−9.81e3 − 8.23e3i)14-s + (−38.9 + 32.6i)15-s + (711. − 4.03e3i)16-s + (1.91e4 − 6.97e3i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.0142 + 0.0807i)3-s + (0.383 − 0.321i)4-s + (0.0363 + 0.0305i)5-s + (0.0100 + 0.0570i)6-s + (−0.882 − 1.52i)7-s + (0.176 − 0.306i)8-s + (0.933 + 0.339i)9-s + (0.0315 + 0.0114i)10-s + (0.160 − 0.277i)11-s + (0.0204 + 0.0354i)12-s + (−0.225 − 1.27i)13-s + (−0.955 − 0.802i)14-s + (−0.00298 + 0.00250i)15-s + (0.0434 − 0.246i)16-s + (0.945 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0443 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.60047 - 1.67302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60047 - 1.67302i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.51 + 2.73i)T \) |
| 19 | \( 1 + (2.71e4 + 1.25e4i)T \) |
| good | 3 | \( 1 + (0.665 - 3.77i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (-10.1 - 8.52i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (800. + 1.38e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-706. + 1.22e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.78e3 + 1.01e4i)T + (-5.89e7 + 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-1.91e4 + 6.97e3i)T + (3.14e8 - 2.63e8i)T^{2} \) |
| 23 | \( 1 + (4.07e4 - 3.42e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-2.30e5 - 8.39e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-4.01e4 - 6.95e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.02e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.37e5 - 7.79e5i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-5.43e5 - 4.56e5i)T + (4.72e10 + 2.67e11i)T^{2} \) |
| 47 | \( 1 + (-5.42e5 - 1.97e5i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 + (-1.06e6 + 8.96e5i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (1.93e6 - 7.04e5i)T + (1.90e12 - 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-1.31e6 + 1.10e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (4.48e5 + 1.63e5i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (4.81e5 + 4.04e5i)T + (1.57e12 + 8.95e12i)T^{2} \) |
| 73 | \( 1 + (-8.30e5 + 4.70e6i)T + (-1.03e13 - 3.77e12i)T^{2} \) |
| 79 | \( 1 + (-7.70e5 + 4.37e6i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (2.91e5 + 5.04e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.72e5 + 1.54e6i)T + (-4.15e13 + 1.51e13i)T^{2} \) |
| 97 | \( 1 + (1.54e6 - 5.62e5i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−14.18042423093148404029057642394, −13.29091868876676638013236199307, −12.36684485026793206690342849055, −10.51014777446686962857647221577, −10.07140254109991364882326259458, −7.70019755973341193178557684971, −6.45079471030617846829460165001, −4.58254048286064484027424212682, −3.23374152355595965611520110857, −0.857486275572834730956727079966,
2.18059042574764777509180761606, 4.03398691309689920078264909791, 5.82009817119682461129611370917, 6.90459838454905452665204661405, 8.780954232237934453954703655507, 10.04008067142047487452779592292, 12.22901172195125372334690867369, 12.31632777005621453092030854079, 13.93297935585760689817873878164, 15.20642691788980957867802608699
