L(s) = 1 | + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (0.719 + 0.694i)5-s + (−0.374 + 0.927i)7-s + (−0.669 − 0.743i)8-s + (−0.5 − 0.866i)10-s + (0.559 + 0.829i)13-s + (0.615 − 0.788i)14-s + (0.438 + 0.898i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.241 + 0.970i)20-s + (−0.766 + 0.642i)23-s + (0.0348 + 0.999i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
L(s) = 1 | + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (0.719 + 0.694i)5-s + (−0.374 + 0.927i)7-s + (−0.669 − 0.743i)8-s + (−0.5 − 0.866i)10-s + (0.559 + 0.829i)13-s + (0.615 − 0.788i)14-s + (0.438 + 0.898i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.241 + 0.970i)20-s + (−0.766 + 0.642i)23-s + (0.0348 + 0.999i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2503678461 + 0.8177718605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2503678461 + 0.8177718605i\) |
\(L(1)\) |
\(\approx\) |
\(0.6826578041 + 0.2274301354i\) |
\(L(1)\) |
\(\approx\) |
\(0.6826578041 + 0.2274301354i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.961 - 0.275i)T \) |
| 5 | \( 1 + (0.719 + 0.694i)T \) |
| 7 | \( 1 + (-0.374 + 0.927i)T \) |
| 13 | \( 1 + (0.559 + 0.829i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.615 + 0.788i)T \) |
| 31 | \( 1 + (-0.997 + 0.0697i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 41 | \( 1 + (0.615 - 0.788i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.848 + 0.529i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.882 - 0.469i)T \) |
| 61 | \( 1 + (-0.997 - 0.0697i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.913 - 0.406i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.961 + 0.275i)T \) |
| 83 | \( 1 + (-0.559 + 0.829i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.719 + 0.694i)T \) |
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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
−24.871724377823233677362325848336, −24.22300145180757475425365086058, −23.27807965333218069000621165709, −22.06964310412968228056934637215, −20.850604375422845131860293942350, −20.120845349758834183548463645039, −19.59031720135383537786124176008, −18.00589608117001893744701273788, −17.73029926488185287275886452440, −16.58294313169718762733531991544, −16.06842168855906518548135997935, −14.88794198212843389546406154771, −13.6274591536518793891364199833, −12.922204908217279542782033547105, −11.47523189570330811876993999145, −10.432127777399665935992699337, −9.74953601193549215422961122748, −8.733013216053249037217981598041, −7.83359232868605891573143213466, −6.61757846831458999339846736769, −5.813581199262085579725868725, −4.4521695808222472006433884651, −2.76588377928609610074368090881, −1.367031714289391399453591243587, −0.352694657319210031769258404745,
1.63472024570804977943490676692, 2.51408307361474410831387721518, 3.65037835845101771757099399456, 5.66967119307600659693099685424, 6.49342603206189642860929999510, 7.4814341709225935616402032622, 8.91437676138931777746798053126, 9.39752584514155811533969715452, 10.47743529394286321452527280279, 11.396535279962752507446163806812, 12.29122491469354589378976860768, 13.50258002577485842096285539801, 14.60591173902054559687439193580, 15.78210080262197306461013802827, 16.39965734682477501780384185244, 17.780801389336760307829410210738, 18.22639474800325285296311695226, 18.99184994998761775362060615624, 19.96773379288698009584699055176, 21.09241057052991624531551524653, 21.78639069649116183702458685667, 22.52234367234637043883916809788, 24.01886153413662770871338262840, 25.0578557552881714400212458528, 25.63497017457148460921726270992