L(s) = 1 | + (−2.75 + 8.47i)2-s + (39.3 + 28.5i)4-s + (42.2 + 130. i)5-s + (29.1 + 21.1i)7-s + (−1.27e3 + 924. i)8-s − 1.21e3·10-s + (−4.38e3 + 512. i)11-s + (4.14e3 − 1.27e4i)13-s + (−259. + 188. i)14-s + (−2.41e3 − 7.42e3i)16-s + (9.61e3 + 2.96e4i)17-s + (−3.89e4 + 2.82e4i)19-s + (−2.05e3 + 6.31e3i)20-s + (7.72e3 − 3.85e4i)22-s − 7.65e4·23-s + ⋯ |
L(s) = 1 | + (−0.243 + 0.749i)2-s + (0.307 + 0.223i)4-s + (0.151 + 0.465i)5-s + (0.0321 + 0.0233i)7-s + (−0.879 + 0.638i)8-s − 0.385·10-s + (−0.993 + 0.116i)11-s + (0.523 − 1.61i)13-s + (−0.0253 + 0.0183i)14-s + (−0.147 − 0.453i)16-s + (0.474 + 1.46i)17-s + (−1.30 + 0.945i)19-s + (−0.0573 + 0.176i)20-s + (0.154 − 0.772i)22-s − 1.31·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.205189 - 0.332925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.205189 - 0.332925i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (4.38e3 - 512. i)T \) |
good | 2 | \( 1 + (2.75 - 8.47i)T + (-103. - 75.2i)T^{2} \) |
| 5 | \( 1 + (-42.2 - 130. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-29.1 - 21.1i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 13 | \( 1 + (-4.14e3 + 1.27e4i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-9.61e3 - 2.96e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (3.89e4 - 2.82e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + 7.65e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.15e5 + 8.40e4i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 31 | \( 1 + (7.01e3 - 2.15e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.12e5 - 1.54e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-4.78e5 + 3.47e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + 7.50e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (8.57e4 - 6.22e4i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-2.81e5 + 8.66e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.04e6 + 7.60e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (3.76e5 + 1.15e6i)T + (-2.54e12 + 1.84e12i)T^{2} \) |
| 67 | \( 1 + 1.79e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (3.42e5 + 1.05e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.87e5 - 1.36e5i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.18e6 - 3.63e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-7.55e5 - 2.32e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + 1.91e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.45e6 - 1.37e7i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−13.10674324859295250243956616799, −12.33596835156448469031751608493, −10.81741701173625475371965299623, −10.17861464031937908339014235140, −8.166197867632191122194505561848, −8.013095533196736192420497699312, −6.36032064801314106655326356629, −5.64848780652032640261275204085, −3.56016773789069093176057034884, −2.18730093599015054022229375230,
0.11931192300316837597377664044, 1.58465468825663445830406314485, 2.76766696532388374878172022464, 4.50015633174616315477739612108, 5.97341304025736089564091055945, 7.22392868739180995004121654017, 8.850826705586849187628241650182, 9.636949349323407583018965032141, 10.87552851890631918858277875659, 11.56896562565236052074142363167