L(s) = 1 | + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)6-s + (0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.669 − 0.743i)14-s + (−0.978 − 0.207i)16-s + (−0.309 + 0.951i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯ |
L(s) = 1 | + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)6-s + (0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.669 − 0.743i)14-s + (−0.978 − 0.207i)16-s + (−0.309 + 0.951i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01324619269 + 0.1672504856i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01324619269 + 0.1672504856i\) |
\(L(1)\) |
\(\approx\) |
\(0.5426919097 - 0.04026137764i\) |
\(L(1)\) |
\(\approx\) |
\(0.5426919097 - 0.04026137764i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 7 | \( 1 + (0.104 + 0.994i)T \) |
| 11 | \( 1 + (0.669 + 0.743i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.809 + 0.587i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (0.104 + 0.994i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (-0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.669 + 0.743i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.669 - 0.743i)T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.913 + 0.406i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.669 + 0.743i)T \) |
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
−17.280550085652717251951323048114, −16.72472393426553167729452337598, −16.27667688611320901512427676368, −15.74688885719790185958928403666, −14.831077143997490066427157065760, −14.40705683275501036872129725988, −13.49227864906076778158348349442, −12.97723791292214812244917394683, −11.83467722372751238428949285222, −11.19029412742145933361058735694, −10.69568316790421522111814293724, −10.16629107818719781642943847593, −9.40009846595970498814601644464, −8.79188212369724079646519821312, −7.88326665161445880332687068927, −7.272456367110827835156428092375, −6.58065764933680075409901680288, −5.93028965006803030732948113312, −5.412419735845945852780463480, −4.48244847801194850193086636017, −3.96367713726973799220456771078, −2.95913318956690268464998285877, −1.44467014874329850455208269257, −0.93061946612543592734713757880, −0.073809716869630197296425769447,
1.46843639740181274426633315049, 1.59827302446279366903345929731, 2.52726634233644211075250855032, 3.4946628585668236871928561012, 4.46808397058711952066461801040, 4.902707895648571805335761480347, 6.02365467339463612838883492958, 6.64985299522674804976544088358, 7.207807053648436270111930330902, 8.1117701534038349159480902070, 8.87738517786078733462884579471, 9.33949695986625497409691070729, 10.106413925404132493880245836512, 11.08266117958830599448160505503, 11.26217957614920192139360476737, 12.09672518338607551877964041742, 12.3998487187251094653053499824, 13.22603679301748790114590656821, 13.724856788511536436621993341861, 14.99318783873708715126429416937, 15.48838259871179930397430664463, 16.306970009522883598850725931747, 17.00387797017473107355520176841, 17.52411374704857613194695209023, 17.89754852712486288435013345381