| L(s) = 1 | + (−6.47 − 4.70i)2-s + (11.4 − 8.33i)3-s + (19.7 + 60.8i)4-s + 363.·5-s − 113.·6-s + (367. + 1.13e3i)7-s + (158. − 486. i)8-s + (−613. + 1.88e3i)9-s + (−2.34e3 − 1.70e3i)10-s + (−908. − 2.79e3i)11-s + (733. + 533. i)12-s + (−5.59e3 + 4.06e3i)13-s + (2.93e3 − 9.04e3i)14-s + (4.16e3 − 3.02e3i)15-s + (−3.31e3 + 2.40e3i)16-s + (−7.30e3 + 2.24e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.245 − 0.178i)3-s + (0.154 + 0.475i)4-s + 1.29·5-s − 0.214·6-s + (0.404 + 1.24i)7-s + (0.109 − 0.336i)8-s + (−0.280 + 0.863i)9-s + (−0.743 − 0.539i)10-s + (−0.205 − 0.633i)11-s + (0.122 + 0.0890i)12-s + (−0.705 + 0.512i)13-s + (0.286 − 0.880i)14-s + (0.318 − 0.231i)15-s + (−0.202 + 0.146i)16-s + (−0.360 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.28592 + 0.866787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28592 + 0.866787i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.47 + 4.70i)T \) |
| 31 | \( 1 + (-1.65e5 - 9.21e3i)T \) |
| good | 3 | \( 1 + (-11.4 + 8.33i)T + (675. - 2.07e3i)T^{2} \) |
| 5 | \( 1 - 363.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-367. - 1.13e3i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 11 | \( 1 + (908. + 2.79e3i)T + (-1.57e7 + 1.14e7i)T^{2} \) |
| 13 | \( 1 + (5.59e3 - 4.06e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (7.30e3 - 2.24e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (3.61e4 + 2.62e4i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 + (1.59e4 - 4.89e4i)T + (-2.75e9 - 2.00e9i)T^{2} \) |
| 29 | \( 1 + (-1.35e5 - 9.82e4i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 37 | \( 1 + 4.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.13e5 - 8.24e4i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + (-6.29e5 - 4.57e5i)T + (8.39e10 + 2.58e11i)T^{2} \) |
| 47 | \( 1 + (4.98e5 - 3.61e5i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-6.09e5 + 1.87e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.81e6 + 1.31e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + 7.27e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.08e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (6.54e5 - 2.01e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-8.73e4 - 2.68e5i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-8.04e5 + 2.47e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-3.89e6 - 2.82e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + (-9.73e5 - 2.99e6i)T + (-3.57e13 + 2.59e13i)T^{2} \) |
| 97 | \( 1 + (-2.43e6 - 7.50e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−13.62248521815441524705113487224, −12.65152491967601338014156052029, −11.32059474596189709083889813498, −10.25146512031011212759554752479, −8.988558571833103137385325889623, −8.276963499474984242103127178354, −6.39765355164052796710959699764, −5.10705847556624763492037395990, −2.54288111999276882880927174931, −1.84721632043535515195365860025,
0.63107024113112924327867392395, 2.31787865978951240189532791510, 4.53805842997837882073814508851, 6.12749999288115017940343713300, 7.27475108222542228417811414108, 8.697130692262334566860093958290, 9.985285883636509556398260642324, 10.39013849436051408107791515546, 12.23895246667323168706294889857, 13.73363187456858592616782686647
