L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s − 8-s + (−0.866 + 0.5i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s − i·31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s − 8-s + (−0.866 + 0.5i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s − i·31-s + (0.5 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00863 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00863 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3294295296 - 0.3322875593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3294295296 - 0.3322875593i\) |
\(L(1)\) |
\(\approx\) |
\(0.9027081645 + 0.3215993266i\) |
\(L(1)\) |
\(\approx\) |
\(0.9027081645 + 0.3215993266i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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
−27.24728081599262011261521938977, −26.22007153017071291257324066770, −24.83683186925003735285019524203, −24.00695884592199487951435722152, −23.17116429424203191717398334414, −21.894934796366689862124223798653, −21.47250454124569419122906961636, −20.46235270774269883254528702568, −19.42932994301707826913574326251, −18.46001991658395976527863017253, −17.81420094682989775612474881371, −16.1481940114618084453802098449, −15.08217575139324043539494863068, −14.271402660239271668454489998311, −13.04467622719210382871060460415, −12.32438032652137297280765556613, −11.1495279665646958689384133381, −10.44774507812201059034678155107, −9.046702063866169756720520073104, −8.18518235568484056996852708764, −6.26410014521358779163335078163, −5.3268802294272041899551224283, −4.19147573915866021593422781122, −2.76795892921327012166143255797, −1.76396995403028221998956062453,
0.12691157149978926865781005243, 2.382806578029663504123908343931, 4.04865969707339846809660906754, 4.81057828818188188754655861131, 6.125224240516520728675921410956, 7.32339854782730712802154019681, 8.00824096331011527353318737858, 9.33853251776759295560426381716, 10.67605459065479512753507463631, 11.87258945183098307438741790977, 13.20828084842534165038742228959, 13.705544052090871191302655951796, 14.981969095201273885510277294939, 15.662893962486128315274266614001, 16.8814961708247184509830820793, 17.58781838116823755772848753644, 18.52194003662717627627822418616, 20.10772255331826664950992870488, 20.90928028363732221465955738178, 21.95281077714087960531695320763, 22.976261513960077628018674037103, 23.75604000091998849501766919024, 24.41430488296137161131087463054, 25.65901426479690486422532311122, 26.31896120252699071521569114034