| L(s) = 1 | + (20.2 + 35.0i)3-s + (−90.2 − 156. i)5-s + 231.·7-s + (274. − 475. i)9-s − 4.27e3·11-s + (−3.99e3 + 6.91e3i)13-s + (3.65e3 − 6.32e3i)15-s + (−1.54e4 − 2.67e4i)17-s + (−2.91e4 + 6.68e3i)19-s + (4.68e3 + 8.11e3i)21-s + (2.41e3 − 4.17e3i)23-s + (2.27e4 − 3.94e4i)25-s + 1.10e5·27-s + (9.75e4 − 1.68e5i)29-s − 2.40e5·31-s + ⋯ |
| L(s) = 1 | + (0.432 + 0.749i)3-s + (−0.322 − 0.559i)5-s + 0.255·7-s + (0.125 − 0.217i)9-s − 0.968·11-s + (−0.504 + 0.873i)13-s + (0.279 − 0.484i)15-s + (−0.763 − 1.32i)17-s + (−0.974 + 0.223i)19-s + (0.110 + 0.191i)21-s + (0.0413 − 0.0716i)23-s + (0.291 − 0.504i)25-s + 1.08·27-s + (0.742 − 1.28i)29-s − 1.45·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.387808 - 0.661891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.387808 - 0.661891i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.91e4 - 6.68e3i)T \) |
| good | 3 | \( 1 + (-20.2 - 35.0i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (90.2 + 156. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 - 231.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.27e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.99e3 - 6.91e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.54e4 + 2.67e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-2.41e3 + 4.17e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-9.75e4 + 1.68e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.88e5 + 5.00e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (3.95e5 + 6.85e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.22e5 - 5.59e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.84e5 + 4.92e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-9.17e5 - 1.58e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.51e5 + 7.82e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.52e5 - 1.64e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.01e6 + 1.75e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.40e6 - 4.16e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.10e6 - 1.91e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 3.20e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.24e6 - 5.62e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-6.46e5 - 1.11e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.69168017184200284566834584801, −11.60343069077273397159399842623, −10.32021625447945823892844538324, −9.259355495583575346954414051322, −8.337318732462912044862724452496, −6.89870497048969203728428603179, −4.99647149093067201485735799144, −4.10867768306426310031432535097, −2.38775347326678275497679698097, −0.23010889285931899824619051109,
1.79698541308469792152911718642, 3.07741498677393727478716912592, 4.92722829201573044459598140713, 6.62473779785212946348158266783, 7.71662259291174813109096181466, 8.495640783896619333695438660956, 10.34384387463525421339251673705, 11.06891445423621388849371275002, 12.81570276970959753082351076449, 13.07355046735245303883722817887
