L(s) = 1 | + (0.374 + 11.3i)2-s + (19.9 − 8.26i)3-s + (−127. + 8.47i)4-s + (−34.6 + 83.5i)5-s + (100. + 222. i)6-s + (−609. − 609. i)7-s + (−143. − 1.44e3i)8-s + (−1.21e3 + 1.21e3i)9-s + (−958. − 360. i)10-s + (−3.76e3 − 1.56e3i)11-s + (−2.47e3 + 1.22e3i)12-s + (−450. − 1.08e3i)13-s + (6.66e3 − 7.12e3i)14-s + 1.95e3i·15-s + (1.62e4 − 2.16e3i)16-s − 1.74e4i·17-s + ⋯ |
L(s) = 1 | + (0.0331 + 0.999i)2-s + (0.426 − 0.176i)3-s + (−0.997 + 0.0662i)4-s + (−0.123 + 0.299i)5-s + (0.190 + 0.420i)6-s + (−0.671 − 0.671i)7-s + (−0.0992 − 0.995i)8-s + (−0.556 + 0.556i)9-s + (−0.302 − 0.113i)10-s + (−0.853 − 0.353i)11-s + (−0.414 + 0.204i)12-s + (−0.0569 − 0.137i)13-s + (0.649 − 0.693i)14-s + 0.149i·15-s + (0.991 − 0.132i)16-s − 0.860i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.0484131 - 0.0749174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0484131 - 0.0749174i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.374 - 11.3i)T \) |
good | 3 | \( 1 + (-19.9 + 8.26i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (34.6 - 83.5i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (609. + 609. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (3.76e3 + 1.56e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (450. + 1.08e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 1.74e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.03e4 + 2.50e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (2.30e4 - 2.30e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.63e5 - 6.78e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 7.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (6.54e4 - 1.58e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (4.54e5 - 4.54e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (-4.74e5 - 1.96e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 4.05e3iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (3.14e5 + 1.30e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-8.03e5 + 1.93e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (1.88e6 - 7.79e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-3.02e6 + 1.25e6i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-6.25e5 - 6.25e5i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.17e6 + 1.17e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 + 6.82e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-2.54e6 - 6.13e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-6.78e5 - 6.78e5i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 8.54e6T + 8.07e13T^{2} \) |
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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
−14.90491433000816066292580343505, −13.70689087385877457673775256902, −13.06000278284622933021626820389, −10.90649505818037209550780484355, −9.388750872738840867123831707191, −7.975264838424194158399994354267, −6.92699374353096033327089556876, −5.22134010301038943801851345025, −3.21402121935425377345870425129, −0.03745625404031204621533010066,
2.34174041540394738079156086363, 3.83417799108647315822766168839, 5.73184829281500559227605991640, 8.297460457153348293969350642248, 9.321576213266817449173477281043, 10.53542415045562046591527462524, 12.14636623387709145307357149810, 12.84598513467839426826774683680, 14.31938141433951300471275662955, 15.41304528105806832221376776748