| L(s) = 1 | + (−4 + 6.92i)2-s + (4.68 + 8.11i)3-s + (−31.9 − 55.4i)4-s + (−231. + 400. i)5-s − 74.9·6-s + 511.·8-s + (1.04e3 − 1.81e3i)9-s + (−1.84e3 − 3.20e3i)10-s + (−1.94e3 − 3.36e3i)11-s + (299. − 519. i)12-s − 1.15e4·13-s − 4.33e3·15-s + (−2.04e3 + 3.54e3i)16-s + (7.62e3 + 1.32e4i)17-s + (8.39e3 + 1.45e4i)18-s + (1.62e4 − 2.81e4i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.100 + 0.173i)3-s + (−0.249 − 0.433i)4-s + (−0.827 + 1.43i)5-s − 0.141·6-s + 0.353·8-s + (0.479 − 0.831i)9-s + (−0.584 − 1.01i)10-s + (−0.439 − 0.761i)11-s + (0.0501 − 0.0867i)12-s − 1.46·13-s − 0.331·15-s + (−0.125 + 0.216i)16-s + (0.376 + 0.651i)17-s + (0.339 + 0.587i)18-s + (0.542 − 0.940i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.864086 - 0.0548350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864086 - 0.0548350i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-4.68 - 8.11i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (231. - 400. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.94e3 + 3.36e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 1.15e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-7.62e3 - 1.32e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.62e4 + 2.81e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.80e4 - 4.86e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 2.64e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.27e4 - 3.93e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.77e5 + 4.80e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.06e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.65e5 + 4.60e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.81e5 - 3.14e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.07e6 - 1.85e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.44e5 - 7.69e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.17e6 + 3.76e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 6.63e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (6.71e5 + 1.16e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.52e6 - 6.11e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 6.60e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.66e6 + 4.61e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 2.09e6T + 8.07e13T^{2} \) |
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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
−12.39212035867634189811864479334, −11.23800036683423449732303578571, −10.28337971043479263571556081359, −9.246821161433026908329824860195, −7.66688274925949882420240175691, −7.14895525017125051768709069536, −5.83154775452800769830032292496, −4.05319807425345745389242053830, −2.78990520551121947236384078248, −0.38447405291136613089518949264,
0.963598637797603295393716499565, 2.38490902560168958067536539483, 4.33151333294144478251675653746, 5.05930783939155074522190022083, 7.55647891009732990948981792482, 7.978214619157010066600439030741, 9.393674198447382549356610151520, 10.23188717106942062397646279658, 11.79037775306629179246857267032, 12.41537742882715797052258271095
