| L(s) = 1 | − i·7-s + i·11-s + 13-s + i·19-s − i·23-s + i·29-s − 31-s − 37-s − 41-s − 43-s + i·47-s − 49-s + 53-s + i·59-s + i·61-s + ⋯ |
| L(s) = 1 | − i·7-s + i·11-s + 13-s + i·19-s − i·23-s + i·29-s − 31-s − 37-s − 41-s − 43-s + i·47-s − 49-s + 53-s + i·59-s + i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3377333475 + 0.6597407274i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3377333475 + 0.6597407274i\) |
| \(L(1)\) |
\(\approx\) |
\(0.9311442606 + 0.06044369198i\) |
| \(L(1)\) |
\(\approx\) |
\(0.9311442606 + 0.06044369198i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 \) |
| 67 | \( 1 \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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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
−18.28655073234991417786165584942, −17.63608206076129696274531896788, −16.78962958351391325461421337427, −16.10737177357135986589067509555, −15.42393676572124671852791262814, −15.077434791048726156530320145428, −13.93937096299458502593783512113, −13.484018105842824777619713782402, −12.86643356163454127395573125492, −11.78344199895915194116131059937, −11.50648074566954916439716856163, −10.75361080903734378340121635320, −9.869813334480288871652151134455, −8.95359967042778531887756772534, −8.674021508698939358938059800139, −7.87630438043992538921211977597, −6.890973253004880553220917001348, −6.159511601825719320179800154880, −5.52572438541903720709515874013, −4.9078166691457195690996053455, −3.65410881700196984780512379458, −3.26174421681580316229599634828, −2.22470902762978350359260324179, −1.451651212032378791188873606128, −0.20102113769374445694487621891,
1.258239385719161428946437690372, 1.74697216118511689213822295432, 2.979033833131960695080898633365, 3.79360683070643675210517828859, 4.34165300342414593010150627336, 5.211605090617534010589835051202, 6.07403557601610661745701969941, 6.95040242199610611154795764376, 7.32937780770826034841596605100, 8.32432110167371650859239536594, 8.86375584288860080073499392131, 9.9291576183851879910020023908, 10.38864653382633192813996022351, 10.962868820893889327601475696102, 11.90765178699520085185607227927, 12.57631812714769468534207313965, 13.23248940982314847355192391658, 13.92797691946806453398859901903, 14.60478671593738750827465342182, 15.1814807845942311174242595694, 16.20945899128333432872287759329, 16.54829766443054682422454945978, 17.28166448328843799331577118511, 18.167801380943996571542868372810, 18.42214487420314584547869429951
