L(s) = 1 | + (−0.0682 − 0.997i)2-s + (0.334 + 0.942i)3-s + (−0.990 + 0.136i)4-s + (−0.962 + 0.269i)5-s + (0.917 − 0.398i)6-s + (−0.854 − 0.519i)7-s + (0.203 + 0.979i)8-s + (−0.775 + 0.631i)9-s + (0.334 + 0.942i)10-s + (−0.775 + 0.631i)11-s + (−0.460 − 0.887i)12-s + (0.775 + 0.631i)13-s + (−0.460 + 0.887i)14-s + (−0.576 − 0.816i)15-s + (0.962 − 0.269i)16-s + (−0.576 + 0.816i)17-s + ⋯ |
L(s) = 1 | + (−0.0682 − 0.997i)2-s + (0.334 + 0.942i)3-s + (−0.990 + 0.136i)4-s + (−0.962 + 0.269i)5-s + (0.917 − 0.398i)6-s + (−0.854 − 0.519i)7-s + (0.203 + 0.979i)8-s + (−0.775 + 0.631i)9-s + (0.334 + 0.942i)10-s + (−0.775 + 0.631i)11-s + (−0.460 − 0.887i)12-s + (0.775 + 0.631i)13-s + (−0.460 + 0.887i)14-s + (−0.576 − 0.816i)15-s + (0.962 − 0.269i)16-s + (−0.576 + 0.816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2057123038 + 0.9419914813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2057123038 + 0.9419914813i\) |
\(L(1)\) |
\(\approx\) |
\(0.7349187775 + 0.1521270701i\) |
\(L(1)\) |
\(\approx\) |
\(0.7349187775 + 0.1521270701i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.0682 - 0.997i)T \) |
| 3 | \( 1 + (0.334 + 0.942i)T \) |
| 5 | \( 1 + (-0.962 + 0.269i)T \) |
| 7 | \( 1 + (-0.854 - 0.519i)T \) |
| 11 | \( 1 + (-0.775 + 0.631i)T \) |
| 13 | \( 1 + (0.775 + 0.631i)T \) |
| 17 | \( 1 + (-0.576 + 0.816i)T \) |
| 19 | \( 1 + (0.334 + 0.942i)T \) |
| 23 | \( 1 + (0.990 - 0.136i)T \) |
| 29 | \( 1 + (0.775 + 0.631i)T \) |
| 31 | \( 1 + (0.203 + 0.979i)T \) |
| 37 | \( 1 + (0.775 + 0.631i)T \) |
| 41 | \( 1 + (0.917 - 0.398i)T \) |
| 43 | \( 1 + (0.576 + 0.816i)T \) |
| 47 | \( 1 + (0.576 + 0.816i)T \) |
| 53 | \( 1 + (-0.990 + 0.136i)T \) |
| 59 | \( 1 + (-0.990 - 0.136i)T \) |
| 61 | \( 1 + (-0.917 + 0.398i)T \) |
| 67 | \( 1 + (-0.203 + 0.979i)T \) |
| 71 | \( 1 + (-0.0682 + 0.997i)T \) |
| 73 | \( 1 + (-0.990 - 0.136i)T \) |
| 79 | \( 1 + (0.990 + 0.136i)T \) |
| 83 | \( 1 + (-0.917 + 0.398i)T \) |
| 89 | \( 1 + (-0.203 + 0.979i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.231361878062612916794397097931, −20.11189727666868770569875106226, −19.43048813397568168504701819563, −18.6759028925458673393922214021, −18.269042384177376501462324492320, −17.27974320416635424492627498467, −16.2696452442580695316312054721, −15.553618581356103888170256722752, −15.25893402412160987754705185636, −13.91571672050799178207221672028, −13.21110834245716601664571907340, −12.82284202032484269126927722651, −11.752972100392775232950316386241, −10.809458395517835751640597238104, −9.24690736424249636206744325919, −8.87003254711868680755709103136, −7.912957684166894401856428945826, −7.3995098872857780211261279852, −6.42088062837214981633589407071, −5.72694342140388114189023959322, −4.672213266895090693935503891617, −3.38212189711254385171825937832, −2.72525850485237028714238172903, −0.6864163317625774642972898744, −0.35064173114380846598123591315,
1.19682422615329536315996336755, 2.7325909447970735512703947362, 3.30147486105474548633913970501, 4.20554277842862875869930087284, 4.65334080347724249698651344782, 6.07708671655030023664196948056, 7.39442354720734539206879668837, 8.30449034075812767453447386968, 9.05493740724479616449641936605, 9.95465512403014780232009042669, 10.71604436085641868652032228300, 11.06891262955418102983142315742, 12.283562507756016664687246503669, 12.940774224208587593586670715610, 13.938171705899293842931726443519, 14.664892765874885332454721481905, 15.672800418309923735455760231544, 16.21598680207909361476557637119, 17.13647464566327337951406011427, 18.22267285647704299080330568658, 19.130719624293386975399814418205, 19.5842247615330623729989662290, 20.389843653137118014373438145335, 20.90657191191893947987929514167, 21.788200310494086871766738965303