| L(s) = 1 | + (8 + 13.8i)2-s + (26.2 − 45.4i)3-s + (−127. + 221. i)4-s + (918. + 1.59e3i)5-s + 839.·6-s − 4.09e3·8-s + (8.46e3 + 1.46e4i)9-s + (−1.46e4 + 2.54e4i)10-s + (−4.69e3 + 8.12e3i)11-s + (6.71e3 + 1.16e4i)12-s − 1.79e5·13-s + 9.63e4·15-s + (−3.27e4 − 5.67e4i)16-s + (4.89e4 − 8.47e4i)17-s + (−1.35e5 + 2.34e5i)18-s + (2.81e5 + 4.87e5i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.186 − 0.323i)3-s + (−0.249 + 0.433i)4-s + (0.656 + 1.13i)5-s + 0.264·6-s − 0.353·8-s + (0.430 + 0.744i)9-s + (−0.464 + 0.804i)10-s + (−0.0965 + 0.167i)11-s + (0.0934 + 0.161i)12-s − 1.73·13-s + 0.491·15-s + (−0.125 − 0.216i)16-s + (0.142 − 0.246i)17-s + (−0.304 + 0.526i)18-s + (0.495 + 0.857i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.114551 - 1.80509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.114551 - 1.80509i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-26.2 + 45.4i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-918. - 1.59e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (4.69e3 - 8.12e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.79e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-4.89e4 + 8.47e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.81e5 - 4.87e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-4.89e5 - 8.47e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 4.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-3.98e6 + 6.90e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (1.35e6 + 2.35e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 9.00e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.10e7 - 3.65e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.40e7 - 5.89e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (1.71e7 - 2.96e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.29e7 + 1.43e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.21e8 - 2.10e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 9.42e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-6.66e7 + 1.15e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (3.38e8 + 5.86e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 4.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.77e8 - 4.80e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.26e9T + 7.60e17T^{2} \) |
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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
−12.93816322830829094287778023683, −11.75073033421868754460065197168, −10.33162395230397832242763511010, −9.556113301900403968052183583447, −7.68508868613057131397510518469, −7.24430605290234545539935876893, −5.95332898082555041584129335118, −4.75723255860385345961392968191, −3.01341904353913661558346519914, −1.95845035759730515852111245670,
0.38307665517392088923735047424, 1.62408255641413359148298240302, 3.05127886306947498739677852048, 4.59523504397545106780705477622, 5.28956668993597371483952606418, 6.91358399546534280101534851732, 8.682984638095967101298232938009, 9.531024341850601402075130827939, 10.24717957878155869219493535155, 11.84990431716745199865568864456
