L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s − 27-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)39-s − 41-s + 43-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s − 27-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)39-s − 41-s + 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05809133048 - 0.9153980680i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05809133048 - 0.9153980680i\) |
\(L(1)\) |
\(\approx\) |
\(0.8900926342 - 0.4412431784i\) |
\(L(1)\) |
\(\approx\) |
\(0.8900926342 - 0.4412431784i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + T \) |
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
−25.82926342582816807885901503238, −25.11295118126900002540350765627, −24.26746314231267401290168520354, −22.68045598691516491723981732220, −22.446862680990665787641990247402, −21.19475209539868921238042000590, −20.46399277907121886182965781802, −19.695896687669003239006341240030, −18.715048646987580511028645173140, −17.3780782308637807030362613081, −16.67086506901344578322823772547, −15.58850765296363828919417853650, −14.75256375860875241466477594592, −14.10524014397834179450268494326, −12.78837169001649623377675017193, −11.76574184193099904597225922606, −10.56417662461236242238967073936, −9.72832669516286784371302185415, −8.92323197473606560065359803655, −7.75992022100376007413767636802, −6.64796639746106895673319619509, −5.066572148479931390939451589127, −4.3773023622470858244728447537, −3.081831659687347577762238109111, −1.93825044336528651684585608417,
0.24221412567922964586417384031, 1.689040306991070657461028764276, 2.8315516611928324403458473031, 4.033135561035002640097842999158, 5.61319982043608483209465767685, 6.66482860569572772268266202, 7.57606126520442148235359574283, 8.657697341386766024191850882342, 9.42868337257793086965315656607, 10.93451379125583950601786992542, 11.85887746257696272040672494006, 12.9102926552583969019974699333, 13.629358014346431835881629479208, 14.66054346398724133025998618980, 15.41731934247481556135886783129, 16.99529416361081118213507952258, 17.46669386101062912525272382708, 18.77964538758074522506340501682, 19.39957244573529177168554625704, 20.08374562484465478062522028501, 21.34374863720757308217484122130, 22.144817260763371103824255934535, 23.33771458380926883011573400004, 24.28787826705449313880734956206, 24.649899357276753228203404895896