L(s) = 1 | + (−1.40 − 0.811i)2-s + (−2.42 + 4.59i)3-s + (−2.68 − 4.64i)4-s + (7.52 + 2.01i)5-s + (7.13 − 4.49i)6-s + (−6.20 + 1.66i)7-s + 21.6i·8-s + (−15.2 − 22.2i)9-s + (−8.93 − 8.93i)10-s + (−11.2 + 3.00i)11-s + (27.8 − 1.08i)12-s + (10.5 + 18.3i)13-s + (10.0 + 2.70i)14-s + (−27.4 + 29.7i)15-s + (−3.85 + 6.68i)16-s + (38.1 − 58.7i)17-s + ⋯ |
L(s) = 1 | + (−0.496 − 0.286i)2-s + (−0.466 + 0.884i)3-s + (−0.335 − 0.580i)4-s + (0.672 + 0.180i)5-s + (0.485 − 0.305i)6-s + (−0.335 + 0.0898i)7-s + 0.958i·8-s + (−0.565 − 0.824i)9-s + (−0.282 − 0.282i)10-s + (−0.307 + 0.0823i)11-s + (0.670 − 0.0259i)12-s + (0.225 + 0.391i)13-s + (0.192 + 0.0515i)14-s + (−0.473 + 0.511i)15-s + (−0.0602 + 0.104i)16-s + (0.544 − 0.838i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0104 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0104 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.491286 - 0.496425i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.491286 - 0.496425i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.42 - 4.59i)T \) |
| 17 | \( 1 + (-38.1 + 58.7i)T \) |
good | 2 | \( 1 + (1.40 + 0.811i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-7.52 - 2.01i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (6.20 - 1.66i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (11.2 - 3.00i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-10.5 - 18.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 153. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-18.3 + 68.3i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (76.9 + 287. i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (-220. - 59.1i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (235. - 235. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-124. + 465. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (34.8 + 20.1i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (72.5 - 125. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 216. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-127. + 73.8i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-27.1 + 7.28i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (-406. - 703. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (429. - 429. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-600. + 600. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (608. - 163. i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (-242. - 139. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 68.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + (229. + 858. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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
−11.85426943795024495283748784367, −11.02056895945145737433845141869, −10.01344554430596871649428515316, −9.558800319325617132455983530146, −8.575411176708139285821799414411, −6.62919951450656838414998233727, −5.54825800752127026814363616909, −4.55935237555977938611722470740, −2.60143952088147818044149063439, −0.44654294279438890784725852584,
1.42848915380552162100486178296, 3.45418284207542107056716588896, 5.42871386184297325367915314706, 6.40129857709179952787687513091, 7.66117202551791353392483642372, 8.338354264749195243233118839228, 9.643768786464068986109309587029, 10.59561866090206492789700643796, 12.07270299156573288504637928041, 12.80690650729982476618316576684