| L(s) = 1 | + (2.47 + 7.60i)2-s + (−2.70 + 8.33i)3-s + (−51.7 + 37.6i)4-s − 344.·5-s − 70.0·6-s + (−605. + 439. i)7-s + (−414. − 300. i)8-s + (1.70e3 + 1.24e3i)9-s + (−851. − 2.62e3i)10-s + (−1.37e3 + 1.00e3i)11-s + (−173. − 533. i)12-s + (4.69e3 − 1.44e4i)13-s + (−4.84e3 − 3.51e3i)14-s + (932. − 2.86e3i)15-s + (1.26e3 − 3.89e3i)16-s + (1.62e4 + 1.17e4i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.0578 + 0.178i)3-s + (−0.404 + 0.293i)4-s − 1.23·5-s − 0.132·6-s + (−0.666 + 0.484i)7-s + (−0.286 − 0.207i)8-s + (0.780 + 0.567i)9-s + (−0.269 − 0.828i)10-s + (−0.312 + 0.227i)11-s + (−0.0289 − 0.0890i)12-s + (0.593 − 1.82i)13-s + (−0.471 − 0.342i)14-s + (0.0713 − 0.219i)15-s + (0.0772 − 0.237i)16-s + (0.800 + 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.671841 - 0.297653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.671841 - 0.297653i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-2.77e4 + 1.63e5i)T \) |
| good | 3 | \( 1 + (2.70 - 8.33i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 5 | \( 1 + 344.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (605. - 439. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (1.37e3 - 1.00e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-4.69e3 + 1.44e4i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.62e4 - 1.17e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.25e4 + 3.87e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (1.16e4 + 8.48e3i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (1.13e4 + 3.49e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 37 | \( 1 - 3.42e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.35e4 + 1.03e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + (-2.56e4 - 7.88e4i)T + (-2.19e11 + 1.59e11i)T^{2} \) |
| 47 | \( 1 + (1.04e5 - 3.22e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.59e6 + 1.16e6i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-6.89e5 + 2.12e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + 1.42e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-2.04e6 - 1.48e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (3.05e6 - 2.21e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (3.26e6 + 2.37e6i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (9.04e5 + 2.78e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-5.41e6 + 3.93e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-1.30e7 + 9.51e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.11967272042042375083830798236, −12.65607754617799799937925845730, −11.12832359852120632334379957000, −9.880362592466523078441491132282, −8.242482724657754155998132633715, −7.52915532636017369041980218370, −5.94671497526082766307279675617, −4.51690405891003491385346487272, −3.16751444361789654903446410055, −0.28566845360034717268893033912,
1.27219295384543592784040544751, 3.49694731622383943728770959121, 4.28947861988008832048019264952, 6.43007815219769097166834600578, 7.67535981481249079099214849835, 9.203256083713697654362205803636, 10.38967185958781331082561803242, 11.69051524798279872716089901177, 12.30520843059643599728501918288, 13.50856812008505265420802164111
