L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.923 + 0.382i)3-s + i·4-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)12-s + i·13-s + (−0.923 + 0.382i)14-s − 16-s − 18-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 + 0.923i)22-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.923 + 0.382i)3-s + i·4-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)12-s + i·13-s + (−0.923 + 0.382i)14-s − 16-s − 18-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 + 0.923i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1878619976 + 0.2312853121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1878619976 + 0.2312853121i\) |
\(L(1)\) |
\(\approx\) |
\(0.4809171008 - 0.05021143568i\) |
\(L(1)\) |
\(\approx\) |
\(0.4809171008 - 0.05021143568i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.923 + 0.382i)T \) |
| 7 | \( 1 + (0.382 - 0.923i)T \) |
| 11 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + iT \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (-0.923 - 0.382i)T \) |
| 29 | \( 1 + (0.382 + 0.923i)T \) |
| 31 | \( 1 + (-0.923 + 0.382i)T \) |
| 37 | \( 1 + (-0.923 + 0.382i)T \) |
| 41 | \( 1 + (-0.382 + 0.923i)T \) |
| 43 | \( 1 + (-0.707 + 0.707i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (0.382 - 0.923i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 + (0.382 + 0.923i)T \) |
| 79 | \( 1 + (-0.923 - 0.382i)T \) |
| 83 | \( 1 + (0.707 + 0.707i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (-0.382 - 0.923i)T \) |
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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
−29.96083612625974683770547708729, −28.69252135137716775852499350623, −28.14290751331128858501107776558, −27.21701073628826965412565761872, −25.844016246854900445272981748325, −24.79742907487160203758010516383, −24.00408947609661202276198087420, −22.97712378417040217263076579158, −21.892107692100140916204205171921, −20.27894555354672375915318399870, −18.82266520730370156309829561181, −18.03516476421401433303106259211, −17.38456534186120695987301505320, −15.88538543618321303094180207356, −15.307854654603355406482362014721, −13.5819883420827389088827879430, −12.18755152687751658093604978137, −10.944079130629956101932451759725, −9.84203684821214663939938273102, −8.24650388555790742650528458539, −7.25919075136511141366955511074, −5.75859644771345418414426010573, −5.10405808486258639897821286812, −2.05535783971545314584385748347, −0.20717594285348801576894711785,
1.42138943384974648112081806380, 3.5938090885561213342213968469, 4.88455369938022644989795511810, 6.76737320987193677819206777395, 8.06234371271459680084749926930, 9.68741508385222667054882592699, 10.614690702425773311077217668629, 11.45802778738970724609660217015, 12.62305802356221352749990681248, 14.03253496192393207100377111714, 16.07837546363239261166771091056, 16.654243014643153951838501866384, 17.856404308542264686669981956605, 18.63054839976274887036657843304, 20.15950539482900242082183281171, 21.07894674627741285297006271362, 21.95716935058511835136291219121, 23.23018453874684405125348735769, 24.20956599625913887453424478791, 26.07460142779635962469057526176, 26.78559616542454034515236261653, 27.64028340433329489059356372367, 28.84395667719613496436849453171, 29.317735470929441484965305746465, 30.482180397220817326528941165929