| L(s) = 1 | + (−10.9 − 2.76i)2-s + (−32.5 − 33.6i)3-s + (112. + 60.7i)4-s + (80.0 + 138. i)5-s + (263. + 458. i)6-s + (1.43e3 + 829. i)7-s + (−1.06e3 − 978. i)8-s + (−71.9 + 2.18e3i)9-s + (−494. − 1.74e3i)10-s + (−5.09e3 − 2.94e3i)11-s + (−1.62e3 − 5.76e3i)12-s + (3.70e3 − 2.13e3i)13-s + (−1.34e4 − 1.30e4i)14-s + (2.05e3 − 7.20e3i)15-s + (9.00e3 + 1.36e4i)16-s + 9.86e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.969 − 0.244i)2-s + (−0.695 − 0.718i)3-s + (0.880 + 0.474i)4-s + (0.286 + 0.496i)5-s + (0.498 + 0.866i)6-s + (1.58 + 0.914i)7-s + (−0.737 − 0.675i)8-s + (−0.0329 + 0.999i)9-s + (−0.156 − 0.551i)10-s + (−1.15 − 0.666i)11-s + (−0.271 − 0.962i)12-s + (0.467 − 0.270i)13-s + (−1.31 − 1.27i)14-s + (0.157 − 0.550i)15-s + (0.549 + 0.835i)16-s + 0.487i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.417441 + 0.460598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.417441 + 0.460598i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.9 + 2.76i)T \) |
| 3 | \( 1 + (32.5 + 33.6i)T \) |
| good | 5 | \( 1 + (-80.0 - 138. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-1.43e3 - 829. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (5.09e3 + 2.94e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-3.70e3 + 2.13e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 9.86e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 4.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-1.15e4 - 2.00e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.39e4 - 4.14e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.49e5 - 8.62e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 5.60e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + (2.80e5 - 1.61e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.62e5 + 2.81e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-1.26e5 + 2.18e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.87e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (6.19e5 - 3.57e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.44e6 - 8.35e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.83e5 - 6.64e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 3.84e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (8.32e5 + 4.80e5i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.82e6 + 1.05e6i)T + (1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 7.79e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (3.08e6 - 5.33e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.17062548711241097314923526231, −12.07235680191207332329672573674, −10.98115165243453168523764672191, −10.61817877037262598190293860304, −8.512670692536501770197114718752, −7.985163734916456214771598206479, −6.46501930337454527221095549269, −5.31108668630010967179591842921, −2.53132389237856298468012927552, −1.46186818512001646460269557187,
0.32444703896950400985056572344, 1.78534070638185829737466788742, 4.48908032899085932605584594524, 5.51091557270726939805398341719, 7.16721364770534924437517258745, 8.333283047275469999684858893252, 9.545739316621664514117166540003, 10.83832922382992706669529955964, 11.05892146976272208036433859694, 12.67274601899037663678258906751
