L(s) = 1 | + (−1.73 + i)2-s + (−5.04 + 1.25i)3-s + (1.99 − 3.46i)4-s + (0.984 − 1.70i)5-s + (7.47 − 7.21i)6-s + (−17.4 − 6.07i)7-s + 7.99i·8-s + (23.8 − 12.6i)9-s + 3.93i·10-s + (15.6 − 9.06i)11-s + (−5.72 + 19.9i)12-s + (7.54 + 4.35i)13-s + (36.3 − 6.97i)14-s + (−2.81 + 9.83i)15-s + (−8 − 13.8i)16-s + 24.9·17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.970 + 0.242i)3-s + (0.249 − 0.433i)4-s + (0.0880 − 0.152i)5-s + (0.508 − 0.491i)6-s + (−0.944 − 0.328i)7-s + 0.353i·8-s + (0.882 − 0.469i)9-s + 0.124i·10-s + (0.430 − 0.248i)11-s + (−0.137 + 0.480i)12-s + (0.161 + 0.0929i)13-s + (0.694 − 0.133i)14-s + (−0.0485 + 0.169i)15-s + (−0.125 − 0.216i)16-s + 0.356·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.705765 + 0.320612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.705765 + 0.320612i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 - i)T \) |
| 3 | \( 1 + (5.04 - 1.25i)T \) |
| 7 | \( 1 + (17.4 + 6.07i)T \) |
good | 5 | \( 1 + (-0.984 + 1.70i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-15.6 + 9.06i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-7.54 - 4.35i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 147. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-133. - 77.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-171. + 99.2i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (18.9 + 10.9i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 4.61T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-209. + 363. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (161. + 279. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-102. - 176. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 512. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-296. + 513. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-294. + 169. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (268. - 465. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 13.3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 27.3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-230. - 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-43.2 - 74.8i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-155. + 89.6i)T + (4.56e5 - 7.90e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84659724330180915055675928215, −11.91751870272027113026119633011, −10.75011499832251906330834325822, −9.942981214236579071553440543185, −9.020097927335630818903437966793, −7.43284881672964623931290649399, −6.39753601348233565945982928083, −5.46201038736323876747235059627, −3.73258928600709825043205874877, −1.02261962201238042103764637272,
0.75741439832996271712166512620, 2.81273763455171669723530530053, 4.77035338855002059684519524255, 6.40283915709087398099155876686, 7.02999101495129925068252649784, 8.700388493401676814149361105215, 9.772470618995101629555533885320, 10.70987782481467950148318142552, 11.64388347777562532397556394084, 12.60510679702133228816248844559