L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 + 0.951i)5-s + (0.913 − 0.406i)7-s + (−0.309 − 0.951i)8-s + (−0.978 − 0.207i)10-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s + (0.5 + 0.866i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.913 − 0.406i)19-s + (0.104 − 0.994i)20-s + (−0.978 − 0.207i)22-s + (0.809 − 0.587i)23-s + ⋯ |
L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 + 0.951i)5-s + (0.913 − 0.406i)7-s + (−0.309 − 0.951i)8-s + (−0.978 − 0.207i)10-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s + (0.5 + 0.866i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.913 − 0.406i)19-s + (0.104 − 0.994i)20-s + (−0.978 − 0.207i)22-s + (0.809 − 0.587i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.301366497 + 0.2919568893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.301366497 + 0.2919568893i\) |
\(L(1)\) |
\(\approx\) |
\(0.8225119310 + 0.4741127021i\) |
\(L(1)\) |
\(\approx\) |
\(0.8225119310 + 0.4741127021i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 211 | \( 1 \) |
good | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + (0.913 - 0.406i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.913 + 0.406i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.809 - 0.587i)T \) |
| 29 | \( 1 + (0.104 - 0.994i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 41 | \( 1 + (0.978 - 0.207i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.913 + 0.406i)T \) |
| 53 | \( 1 + (0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.669 - 0.743i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.669 - 0.743i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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
−22.48280305688688009136216139815, −21.53528761175579602602245765055, −21.133157573551062368272067435498, −20.22614862173900720028069222628, −19.579519320914725528613115604545, −18.66857977074207884467500998790, −17.909462820565486684091315441731, −17.01008975689863742786990620125, −16.12327669890736779046343407471, −15.00585141554360908188034522654, −14.09858840190504362640662591562, −13.303390500079263873762350771366, −12.46057148593238731263945619352, −11.458088610211048550303892424908, −11.29654255283613936784120385501, −9.85381632380253668752891711805, −8.97534161162996829369818392633, −8.42732215223518889245791710785, −7.36715760770046987186147863217, −5.55361720573615550503988645931, −5.03862085147150911794315375424, −4.14952396169570973459959744280, −2.96857077082590031995996921056, −1.83251550572823791016521844744, −0.885288608627062873065881467969,
0.39110780167116172851594024558, 2.16422815486396214963412791942, 3.43093344733514139287733868707, 4.5735500903396096113026090673, 5.17577123877192182772637412453, 6.530638904161268130889192673613, 7.32002294573402538608555599472, 7.7687252002626018433166817840, 8.889647982304138071470411001866, 10.04667234539824430928759645226, 10.7599786934141999981901289440, 11.89720143458501256608485038943, 12.89072405127599206737030892723, 13.87971943004252124978515347316, 14.65557349049220409156284220979, 15.19272201141099340385984563534, 15.88836490024751692675078591703, 17.27733099854035786891644965785, 17.64593978776348512992987282419, 18.330138136598894748437436361654, 19.38844976217332508845353900460, 20.290962572265054430324595552922, 21.38976039023238349946370082786, 22.30428873511861992740436660185, 22.85575769111765138793922150543