| L(s) = 1 | + (−16.2 + 11.8i)2-s + (85.7 − 263. i)4-s + (21.6 + 15.7i)5-s + (176. − 543. i)7-s + (929. + 2.86e3i)8-s − 539.·10-s + (−631. + 4.36e3i)11-s + (−4.42e3 + 3.21e3i)13-s + (3.55e3 + 1.09e4i)14-s + (−2.02e4 − 1.47e4i)16-s + (6.44e3 + 4.68e3i)17-s + (−4.60e3 − 1.41e4i)19-s + (6.01e3 − 4.36e3i)20-s + (−4.14e4 − 7.86e4i)22-s − 8.62e3·23-s + ⋯ |
| L(s) = 1 | + (−1.43 + 1.04i)2-s + (0.669 − 2.06i)4-s + (0.0775 + 0.0563i)5-s + (0.194 − 0.599i)7-s + (0.642 + 1.97i)8-s − 0.170·10-s + (−0.142 + 0.989i)11-s + (−0.558 + 0.405i)13-s + (0.346 + 1.06i)14-s + (−1.23 − 0.899i)16-s + (0.318 + 0.231i)17-s + (−0.154 − 0.474i)19-s + (0.167 − 0.122i)20-s + (−0.829 − 1.57i)22-s − 0.147·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.727068 + 0.0535096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.727068 + 0.0535096i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (631. - 4.36e3i)T \) |
| good | 2 | \( 1 + (16.2 - 11.8i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-21.6 - 15.7i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-176. + 543. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (4.42e3 - 3.21e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-6.44e3 - 4.68e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (4.60e3 + 1.41e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 8.62e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-4.37e4 + 1.34e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.31e5 - 9.52e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-7.74e4 + 2.38e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-1.12e5 - 3.46e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 6.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-4.78e4 - 1.47e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.00e6 - 7.30e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-7.17e5 + 2.20e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-8.01e5 - 5.82e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 5.95e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (3.62e6 + 2.63e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-1.76e6 + 5.41e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-1.85e6 + 1.34e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-7.03e6 - 5.10e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 2.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-8.90e6 + 6.46e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
| 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
−12.43740974039240665411494658593, −10.93903502145286497870159329986, −10.01650470335369097880013250675, −9.231349693820055196586863937713, −7.912943259014457819631935981428, −7.22196513164300357452816283480, −6.12081589346620998760564437560, −4.56947902700938021538592245489, −1.99829086580092940934890487942, −0.49979567300953040916991786747,
0.869791075105361627412523835573, 2.25260623975700341611247751830, 3.42913230844639575908521775571, 5.56926917679262086112458894347, 7.41510396442075851104714562568, 8.421833681046765840792498624922, 9.259597547393214186156587804957, 10.30104453894853566422394908787, 11.22952972148899065029008498650, 12.08320545344553665101000102836
