| L(s) = 1 | + (4 + 6.92i)2-s + (−39.6 − 68.6i)3-s + (−31.9 + 55.4i)4-s + 132.·5-s + (317. − 549. i)6-s + (−754. + 1.30e3i)7-s − 511.·8-s + (−2.04e3 + 3.54e3i)9-s + (530. + 918. i)10-s + (1.46e3 + 2.53e3i)11-s + 5.07e3·12-s + (−7.91e3 + 240. i)13-s − 1.20e4·14-s + (−5.25e3 − 9.10e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (1.08e4 − 1.88e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.847 − 1.46i)3-s + (−0.249 + 0.433i)4-s + 0.474·5-s + (0.599 − 1.03i)6-s + (−0.831 + 1.43i)7-s − 0.353·8-s + (−0.936 + 1.62i)9-s + (0.167 + 0.290i)10-s + (0.331 + 0.574i)11-s + 0.847·12-s + (−0.999 + 0.0303i)13-s − 1.17·14-s + (−0.402 − 0.696i)15-s + (−0.125 − 0.216i)16-s + (0.536 − 0.929i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.218097 + 0.555195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218097 + 0.555195i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 13 | \( 1 + (7.91e3 - 240. i)T \) |
| good | 3 | \( 1 + (39.6 + 68.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 132.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (754. - 1.30e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.46e3 - 2.53e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.08e4 + 1.88e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.67e4 - 4.63e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.42e4 - 2.46e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.63e4 + 1.32e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 8.33e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (5.62e4 + 9.73e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (5.50e4 + 9.53e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (5.29e4 - 9.17e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 4.13e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.89e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.20e6 - 2.08e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.19e6 - 2.06e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.84e5 - 3.20e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.59e6 + 2.76e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 5.84e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.47e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.81e6 - 1.00e7i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.11e4 - 5.40e4i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−16.59742533803894260550626105551, −15.06492233485527015483765129699, −13.63563467263103993109578490318, −12.35069428524569218379174926674, −12.07615204928541046212082866356, −9.552428130896061108292630343792, −7.68362353717483614009415953166, −6.34734425840558225407562277420, −5.53261499213347158230359298570, −2.20771799778827641035146128298,
0.28637075526289471709435495819, 3.55158826630784322412726203703, 4.79274759649410613750469996864, 6.41202421035417755654813853444, 9.428424844502128801272979713681, 10.32530937664371271082009950382, 11.08982565283734636784068412085, 12.76343221090491441542356669068, 14.14185966469543196943693500860, 15.46323042147239332232149545198
