L(s) = 1 | − 2-s + (0.0581 + 0.998i)3-s + 4-s + (0.918 − 0.396i)5-s + (−0.0581 − 0.998i)6-s + (−0.993 − 0.116i)7-s − 8-s + (−0.993 + 0.116i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (0.0581 + 0.998i)12-s + (−0.396 − 0.918i)13-s + (0.993 + 0.116i)14-s + (0.448 + 0.893i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (0.0581 + 0.998i)3-s + 4-s + (0.918 − 0.396i)5-s + (−0.0581 − 0.998i)6-s + (−0.993 − 0.116i)7-s − 8-s + (−0.993 + 0.116i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (0.0581 + 0.998i)12-s + (−0.396 − 0.918i)13-s + (0.993 + 0.116i)14-s + (0.448 + 0.893i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05929565617 + 0.3515692522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05929565617 + 0.3515692522i\) |
\(L(1)\) |
\(\approx\) |
\(0.6014149653 + 0.2280978190i\) |
\(L(1)\) |
\(\approx\) |
\(0.6014149653 + 0.2280978190i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.0581 + 0.998i)T \) |
| 5 | \( 1 + (0.918 - 0.396i)T \) |
| 7 | \( 1 + (-0.993 - 0.116i)T \) |
| 11 | \( 1 + (0.918 + 0.396i)T \) |
| 13 | \( 1 + (-0.396 - 0.918i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.116 + 0.993i)T \) |
| 31 | \( 1 + (0.727 + 0.686i)T \) |
| 41 | \( 1 + (-0.642 - 0.766i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.998 - 0.0581i)T \) |
| 53 | \( 1 + (-0.549 - 0.835i)T \) |
| 59 | \( 1 + (-0.893 + 0.448i)T \) |
| 61 | \( 1 + (-0.230 - 0.973i)T \) |
| 67 | \( 1 + (-0.998 - 0.0581i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.993 - 0.116i)T \) |
| 79 | \( 1 + (-0.893 + 0.448i)T \) |
| 83 | \( 1 + (0.286 - 0.957i)T \) |
| 89 | \( 1 + (0.802 + 0.597i)T \) |
| 97 | \( 1 + (-0.727 + 0.686i)T \) |
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
−18.228024723577353173339547104708, −17.41954089382272477094383922768, −16.88965032039411682743660915602, −16.51111977103321230008726295381, −15.36977747339272875535453654525, −14.73845742923447648157502634426, −13.830349799013071086543627859506, −13.3870576692018980705312459829, −12.59231109950636922183939392050, −11.57416275080741455722148001006, −11.467008558719982235701271450992, −10.356026266763618786176777228747, −9.487953131057493607065407863276, −9.1800600363337477396865070634, −8.54129525399845508396908103915, −7.40565732709929900850869417683, −6.75025410869613534212740772392, −6.4816400075411000994543120125, −5.90620678989710324384336974620, −4.66564873823195260053374346057, −3.15884345290248913769788530867, −2.714021400582818898355318997857, −2.006001198556013641348898055334, −1.1463552213455189503088812849, −0.14024876002392354216634423883,
1.251048846396935272687955491569, 2.01676042723287748896281774918, 3.13891273205357327073540438415, 3.50950874866714862658664164004, 4.7485488070908229757743740583, 5.550669169782443177484181605769, 6.287900758150487731922296399993, 6.79175149988910424907141922051, 7.94772008560744459513930717391, 8.76860849114487850999097160233, 9.18338429384869723496675985881, 9.93977365841350120470398991389, 10.239981991462249075414215179224, 10.86971899596095675113134596509, 11.9506942790517178331809146473, 12.55411142589176955138254057125, 13.30817373609167777662793018095, 14.3628211969504392403494513391, 14.95029903079568000295397669457, 15.58384176981678481532244669984, 16.3693584224105316926709562431, 16.86827762926285511846539934343, 17.35115364228157413400962407469, 17.86960496547462095218312440876, 18.94165314556919034236540085865