L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.342 − 0.939i)5-s + (−0.173 − 0.984i)6-s + (0.866 − 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + i·10-s + (0.342 + 0.939i)11-s + (−0.173 + 0.984i)12-s + (0.984 − 0.173i)13-s + (−0.984 + 0.173i)14-s + (0.642 − 0.766i)15-s + (0.173 + 0.984i)16-s + (−0.866 − 0.5i)17-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.342 − 0.939i)5-s + (−0.173 − 0.984i)6-s + (0.866 − 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + i·10-s + (0.342 + 0.939i)11-s + (−0.173 + 0.984i)12-s + (0.984 − 0.173i)13-s + (−0.984 + 0.173i)14-s + (0.642 − 0.766i)15-s + (0.173 + 0.984i)16-s + (−0.866 − 0.5i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7663966186 + 0.02355251800i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7663966186 + 0.02355251800i\) |
\(L(1)\) |
\(\approx\) |
\(0.8355444745 + 0.01032304419i\) |
\(L(1)\) |
\(\approx\) |
\(0.8355444745 + 0.01032304419i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.342 - 0.939i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.342 + 0.939i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.342 + 0.939i)T \) |
| 31 | \( 1 + (-0.642 - 0.766i)T \) |
| 37 | \( 1 + (-0.939 + 0.342i)T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (-0.984 - 0.173i)T \) |
| 53 | \( 1 + (0.342 - 0.939i)T \) |
| 59 | \( 1 + (-0.984 + 0.173i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 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
−31.17060189533640387618595094746, −30.44781734412241903372406922114, −29.40570579219387568399429757718, −28.22703456165460358518663597828, −26.90552644354690203727963948102, −26.257703871686222846676596353873, −25.03023311781580645168626086063, −24.25654816166284712562523859428, −23.23352407828415065965842334129, −21.44824083574992677831899538306, −20.06200661970700365591751660876, −19.00697220531294836680012893204, −18.35107679556160553526771080221, −17.45883612446192906736239947576, −15.703266507374664503042721716508, −14.73465933103154708464530841135, −13.70458243269948052348345136398, −11.623052497375575861157215652586, −11.018924522842500418409529486689, −9.025720977661862474603132256170, −8.18091306147684789337970562313, −7.01932316027153988905098817818, −5.9382791408685592145814931549, −3.12183658162530161909760570387, −1.592299349568072614860886335497,
1.62645926424466074053834059009, 3.64857587976075883950468799733, 4.9055965231168360149940450029, 7.3851792137668566839023312199, 8.532538162752212672756832304185, 9.35143703535484790655853692279, 10.70777799134582561155071434624, 11.7135921808415444899847549776, 13.31128519317833549247447258624, 14.988786846512179996140862762929, 16.06566371494271130774966229786, 16.97920682036604592189945738975, 18.13910910971505697179466189708, 19.808630535464056796246405237452, 20.49542268435335141438948182381, 20.9860289824625071745736452199, 22.61541764558229745508294599691, 24.3199796451943100816197070140, 25.27532223408318963126136446540, 26.414240729862731355113013603859, 27.48132323270061688794898535029, 27.89628345894253293498949098781, 29.063033753322010827602206475647, 30.75703357179634360332786780837, 31.177473798505921125510807397501