L(s) = 1 | + (−0.978 + 0.207i)3-s + (0.104 + 0.994i)5-s + (0.913 − 0.406i)9-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)15-s + (0.913 + 0.406i)17-s + (−0.669 − 0.743i)19-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.104 + 0.994i)31-s + (−0.978 − 0.207i)37-s + (0.913 + 0.406i)39-s + (0.309 − 0.951i)41-s + ⋯ |
L(s) = 1 | + (−0.978 + 0.207i)3-s + (0.104 + 0.994i)5-s + (0.913 − 0.406i)9-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)15-s + (0.913 + 0.406i)17-s + (−0.669 − 0.743i)19-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.104 + 0.994i)31-s + (−0.978 − 0.207i)37-s + (0.913 + 0.406i)39-s + (0.309 − 0.951i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.122737954 - 0.1179790034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.122737954 - 0.1179790034i\) |
\(L(1)\) |
\(\approx\) |
\(0.7967236549 + 0.08678004006i\) |
\(L(1)\) |
\(\approx\) |
\(0.7967236549 + 0.08678004006i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.913 + 0.406i)T \) |
| 19 | \( 1 + (-0.669 - 0.743i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (-0.104 + 0.994i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.669 - 0.743i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−24.86361653932953870064299807435, −24.09449266244768171736939684831, −23.42623747724022197140646205403, −22.48880785301150266367572143333, −21.44707267256300606234705460903, −20.84651033071666074089963725766, −19.554458579771328105923893246248, −18.751731108502150910885226430348, −17.650980866148795092031592486485, −16.76635034696294772822886139537, −16.429794194790252931613480739835, −15.17134148136311853227154169773, −13.90952025914528170418725956371, −12.81691286907608908340755256277, −12.197976840548262586175846892, −11.34950654939281378794307990410, −10.09907956637736339695232096423, −9.27977548677412967462146783489, −7.94148603352656004910367227900, −6.97120721465428402332060738773, −5.694085032088793744226757922000, −5.02989849999541822977667055437, −3.92419970492193761691608594502, −1.9716890283068763381052796387, −0.85134803562025485631772674431,
0.5272563158967357399139264480, 2.28861994231209241667296197899, 3.58900416385341643174995751331, 4.86473570192697859104777591262, 5.89642264278229357363019090521, 6.81320971793432604692249825316, 7.70725055277471972666333943925, 9.31684697624272524152533417257, 10.44856685963356516392975304158, 10.80122343179753464571618600167, 12.04957028286729025379001964401, 12.78645105467671056731814858370, 14.17630571987960113031198773328, 15.06436935256641468131637425064, 15.85249464904205497179130941104, 17.18094567083011443436261662816, 17.523022101527900289391137577183, 18.71319526489748877547224773048, 19.32031101328399706264463392455, 20.80575863495527544765436020008, 21.65918688785498528702164306105, 22.42090249632015634597018790605, 23.04535297023507714338396252213, 23.94948821693260827359055075322, 24.98917258638977812959518417497