L(s) = 1 | + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.988 + 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.365 − 0.930i)10-s + (0.733 + 0.680i)11-s + (−0.733 − 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.826 + 0.563i)26-s + ⋯ |
L(s) = 1 | + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.988 + 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.365 − 0.930i)10-s + (0.733 + 0.680i)11-s + (−0.733 − 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.826 + 0.563i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2074576696 - 1.631911286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2074576696 - 1.631911286i\) |
\(L(1)\) |
\(\approx\) |
\(0.9349462652 - 0.6766011347i\) |
\(L(1)\) |
\(\approx\) |
\(0.9349462652 - 0.6766011347i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.988 + 0.149i)T \) |
| 11 | \( 1 + (0.733 + 0.680i)T \) |
| 13 | \( 1 + (-0.733 - 0.680i)T \) |
| 17 | \( 1 + (-0.826 - 0.563i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.0747 - 0.997i)T \) |
| 29 | \( 1 + (-0.826 - 0.563i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.0747 - 0.997i)T \) |
| 41 | \( 1 + (-0.365 - 0.930i)T \) |
| 43 | \( 1 + (0.365 - 0.930i)T \) |
| 47 | \( 1 + (0.222 - 0.974i)T \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T \) |
| 59 | \( 1 + (-0.623 - 0.781i)T \) |
| 61 | \( 1 + (-0.900 + 0.433i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.733 + 0.680i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.733 - 0.680i)T \) |
| 89 | \( 1 + (-0.955 - 0.294i)T \) |
| 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
−24.32277313274404403603882173432, −23.69825906596391634212398958278, −22.37717023859546220986387819821, −21.80754661985293733452897510235, −21.29304139524228616664547659727, −19.83357316411545795474824953331, −18.935963540119081560671406790, −17.85345117083957324529534002816, −17.13571059679602925161838391098, −16.664331324984846278007650198030, −15.463348787529593404872826700165, −14.63565883854684474232032233012, −13.75246800247620356599757116415, −13.21807535424795914495241068572, −12.11495087461263318777418866490, −10.90748570504131244978315796450, −9.477186299625908195421853968882, −9.12710693679998673159958187495, −7.99166506238206420222439585043, −6.681399929972194634581908372360, −6.2384717413700075063201126560, −5.07553549229318449459884493901, −4.21966980515275858591753463708, −2.81808730546581757522474232275, −1.32377625052034750581588361120,
0.40197753434844885971022566274, 1.89057576938339259619043843394, 2.50171369482548149916995980175, 3.8909077968744775704997362178, 4.90329477183872486878776495831, 5.87238048563920663032671480497, 6.95826454873542325066013483220, 8.4693690337018011464900134622, 9.45169302544274598940635425947, 10.094860792270257221491960596964, 10.91060003040461795383291219231, 12.12033382325717128825070660981, 12.733558847353350871167699221928, 13.731409424720943091468406721940, 14.466725727171357618512321070056, 15.25644179321327839963311266076, 16.93444488974380795113816817476, 17.51777454502510011446975204179, 18.34258909768626882989614903658, 19.23047917124777370495354004828, 20.27191245779179376703748572781, 20.746767932013843622940236739993, 21.79480256624637741156863230935, 22.49105491935295265256860998651, 22.958709561748009087884498280195