| L(s) = 1 | + (161. − 279. i)5-s + (−882. + 210. i)7-s + (−43.6 − 75.6i)11-s − 9.60e3·13-s + (1.61e4 + 2.80e4i)17-s + (−477. + 826. i)19-s + (−4.78e3 + 8.27e3i)23-s + (−1.29e4 − 2.24e4i)25-s + 2.10e5·29-s + (2.08e4 + 3.60e4i)31-s + (−8.36e4 + 2.80e5i)35-s + (−1.39e4 + 2.42e4i)37-s − 2.96e5·41-s + 4.96e5·43-s + (3.08e5 − 5.34e5i)47-s + ⋯ |
| L(s) = 1 | + (0.576 − 0.999i)5-s + (−0.972 + 0.231i)7-s + (−0.00989 − 0.0171i)11-s − 1.21·13-s + (0.799 + 1.38i)17-s + (−0.0159 + 0.0276i)19-s + (−0.0819 + 0.141i)23-s + (−0.165 − 0.286i)25-s + 1.59·29-s + (0.125 + 0.217i)31-s + (−0.329 + 1.10i)35-s + (−0.0454 + 0.0786i)37-s − 0.671·41-s + 0.951·43-s + (0.433 − 0.750i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.854400402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854400402\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (882. - 210. i)T \) |
| good | 5 | \( 1 + (-161. + 279. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (43.6 + 75.6i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 9.60e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.61e4 - 2.80e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (477. - 826. i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.78e3 - 8.27e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 2.10e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.08e4 - 3.60e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.39e4 - 2.42e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.08e5 + 5.34e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (2.99e5 + 5.19e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.20e6 + 2.08e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.39e5 - 9.34e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.37e6 + 2.37e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.33e6 - 2.30e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.44e6 + 2.50e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.60e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.66e5 - 8.07e5i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.21e7T + 8.07e13T^{2} \) |
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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
−10.45396497281182387008995023023, −9.784056039374685723753106634669, −8.952689977631597642504796906214, −7.961945687113092709582872190883, −6.63069876506594325809004527856, −5.67513523732200831834724677540, −4.71591100640454228475318236206, −3.31813623188979100071926002956, −1.99293402345879507107371153901, −0.69987957666320816424478386156,
0.65816637014139211596068865218, 2.50664175187161022903565672384, 3.10056783453499012448352477341, 4.68761886947497954568062227773, 5.97071816776675629019078096777, 6.87216461594416216437070089695, 7.59666030682402719267852592617, 9.215532461814195610164492382722, 9.971813035082171296042738182857, 10.53914545716233103840287877691
