| L(s) = 1 | + (−10.8 + 7.85i)2-s + (15.6 − 48.1i)4-s + (386. + 280. i)5-s + (41.4 − 127. i)7-s + (−319. − 983. i)8-s − 6.37e3·10-s + (−2.17e3 − 3.83e3i)11-s + (−5.09e3 + 3.70e3i)13-s + (554. + 1.70e3i)14-s + (1.64e4 + 1.19e4i)16-s + (1.10e4 + 7.99e3i)17-s + (4.80e3 + 1.47e4i)19-s + (1.95e4 − 1.42e4i)20-s + (5.37e4 + 2.43e4i)22-s − 1.11e5·23-s + ⋯ |
| L(s) = 1 | + (−0.955 + 0.694i)2-s + (0.122 − 0.376i)4-s + (1.38 + 1.00i)5-s + (0.0456 − 0.140i)7-s + (−0.220 − 0.679i)8-s − 2.01·10-s + (−0.493 − 0.869i)11-s + (−0.643 + 0.467i)13-s + (0.0539 + 0.166i)14-s + (1.00 + 0.728i)16-s + (0.543 + 0.394i)17-s + (0.160 + 0.494i)19-s + (0.546 − 0.397i)20-s + (1.07 + 0.488i)22-s − 1.91·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.164491 - 0.385485i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.164491 - 0.385485i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (2.17e3 + 3.83e3i)T \) |
| good | 2 | \( 1 + (10.8 - 7.85i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-386. - 280. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-41.4 + 127. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (5.09e3 - 3.70e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.10e4 - 7.99e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-4.80e3 - 1.47e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 1.11e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + (8.22e3 - 2.53e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (7.58e4 - 5.51e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (1.29e5 - 3.97e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.92e5 + 5.91e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 8.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (1.53e5 + 4.70e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-7.28e5 + 5.29e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (4.54e5 - 1.40e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.10e6 + 8.00e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 2.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (2.18e6 + 1.58e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-1.21e6 + 3.73e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (3.61e6 - 2.62e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-2.93e6 - 2.13e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 4.81e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.45e5 - 1.78e5i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.50217911790276869301823115587, −12.04469127515040916561663883538, −10.33695876300426256504291369619, −10.05209149562067545434086017256, −8.774607934591365604697815171425, −7.61315615853502944630236734980, −6.52437006555788688650070126854, −5.66608392895609704053711056114, −3.36276548372785074415000430499, −1.78725452741680939770994062891,
0.16822512014855748615176448104, 1.57705862356703333894769275472, 2.43826753736001612551765846104, 4.92836859408998133168784339665, 5.79850561454672194767588743970, 7.74525969071075282263689333026, 8.910101196789042201766376680497, 9.858307794339482173450071563841, 10.17729484190546892988751787645, 11.81184459204103209504884283264
