L(s) = 1 | + (−0.970 − 0.241i)2-s + (0.999 + 0.0348i)3-s + (0.882 + 0.469i)4-s + (−0.961 − 0.275i)6-s + (−0.743 + 0.669i)7-s + (−0.743 − 0.669i)8-s + (0.997 + 0.0697i)9-s + (0.866 + 0.5i)12-s + (−0.829 − 0.559i)13-s + (0.882 − 0.469i)14-s + (0.559 + 0.829i)16-s + (−0.0697 − 0.997i)17-s + (−0.951 − 0.309i)18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (−0.719 − 0.694i)24-s + ⋯ |
L(s) = 1 | + (−0.970 − 0.241i)2-s + (0.999 + 0.0348i)3-s + (0.882 + 0.469i)4-s + (−0.961 − 0.275i)6-s + (−0.743 + 0.669i)7-s + (−0.743 − 0.669i)8-s + (0.997 + 0.0697i)9-s + (0.866 + 0.5i)12-s + (−0.829 − 0.559i)13-s + (0.882 − 0.469i)14-s + (0.559 + 0.829i)16-s + (−0.0697 − 0.997i)17-s + (−0.951 − 0.309i)18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (−0.719 − 0.694i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.216578459 - 0.3058632802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216578459 - 0.3058632802i\) |
\(L(1)\) |
\(\approx\) |
\(0.9370760327 - 0.09394750394i\) |
\(L(1)\) |
\(\approx\) |
\(0.9370760327 - 0.09394750394i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.970 - 0.241i)T \) |
| 3 | \( 1 + (0.999 + 0.0348i)T \) |
| 7 | \( 1 + (-0.743 + 0.669i)T \) |
| 13 | \( 1 + (-0.829 - 0.559i)T \) |
| 17 | \( 1 + (-0.0697 - 0.997i)T \) |
| 23 | \( 1 + (0.984 + 0.173i)T \) |
| 29 | \( 1 + (0.848 - 0.529i)T \) |
| 31 | \( 1 + (-0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.951 - 0.309i)T \) |
| 41 | \( 1 + (-0.0348 + 0.999i)T \) |
| 43 | \( 1 + (0.984 - 0.173i)T \) |
| 47 | \( 1 + (0.927 + 0.374i)T \) |
| 53 | \( 1 + (-0.898 + 0.438i)T \) |
| 59 | \( 1 + (0.374 + 0.927i)T \) |
| 61 | \( 1 + (0.719 - 0.694i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.438 - 0.898i)T \) |
| 73 | \( 1 + (0.788 + 0.615i)T \) |
| 79 | \( 1 + (0.961 - 0.275i)T \) |
| 83 | \( 1 + (-0.406 - 0.913i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.970 + 0.241i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.29934682414545729067543891535, −20.588065738353125025454394759083, −19.75315795677709490253408620291, −19.28998056086892031227024653079, −18.822335334672626009434518340989, −17.59901289611131083519006208762, −17.01822862849059231205212403227, −16.08921568180527468250644843590, −15.49980588615162546722844807101, −14.52789281490512366518034900692, −14.020086548422123629788269962765, −12.84342020084371110936761836007, −12.193772747782909200866261088269, −10.7678600616348949403326138362, −10.25815734911500401520365338789, −9.40706288023024523847112812514, −8.77809505128140254228021839619, −7.941147102149283216859777262004, −6.90540788221758334680993021509, −6.73972202379000479975927299786, −5.19158827745601450648690194453, −3.94282887024780562317362813918, −3.00551737669781113607170486403, −2.08125178056078533802988759450, −1.011743471372129271667334986192,
0.77077874629457651782772022200, 2.252141449337822193499878379954, 2.744750031501643161722385011386, 3.55780118411381563121575128373, 4.92568060376941158265027431500, 6.225454899144006989174760020931, 7.1758061435049668139713647907, 7.798116989151002289361445885120, 8.76613737069879450141239764642, 9.42562275094612837939514932197, 9.897611214609347652870895409330, 10.89504418705307142841582677123, 12.02393462619837288134453444300, 12.64026386160615930069809156214, 13.47721736898341534461585621768, 14.59717149391911981506427681984, 15.4748687684297284553924745789, 15.82611596633378437172639009929, 16.85783802361652717745657365032, 17.74285062483507907282664391793, 18.64681319486449513577855584074, 19.13304314098891468606314231296, 19.79339630507878074396164764214, 20.5092666021640514625932212872, 21.22439859679951961573962920934