L(s) = 1 | + (0.913 + 0.406i)5-s + (0.309 + 0.951i)7-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)19-s − 23-s + (0.669 + 0.743i)25-s + (−0.669 + 0.743i)29-s + (0.104 − 0.994i)31-s + (−0.104 + 0.994i)35-s + (−0.978 − 0.207i)37-s + (−0.309 + 0.951i)41-s + 43-s + (−0.669 − 0.743i)47-s + (−0.809 + 0.587i)49-s + (−0.809 − 0.587i)53-s + ⋯ |
L(s) = 1 | + (0.913 + 0.406i)5-s + (0.309 + 0.951i)7-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)19-s − 23-s + (0.669 + 0.743i)25-s + (−0.669 + 0.743i)29-s + (0.104 − 0.994i)31-s + (−0.104 + 0.994i)35-s + (−0.978 − 0.207i)37-s + (−0.309 + 0.951i)41-s + 43-s + (−0.669 − 0.743i)47-s + (−0.809 + 0.587i)49-s + (−0.809 − 0.587i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01347940001 + 0.04135306159i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01347940001 + 0.04135306159i\) |
\(L(1)\) |
\(\approx\) |
\(0.9620053682 + 0.1812053613i\) |
\(L(1)\) |
\(\approx\) |
\(0.9620053682 + 0.1812053613i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (0.913 + 0.406i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.669 + 0.743i)T \) |
| 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 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.913 - 0.406i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (-0.104 - 0.994i)T \) |
| 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
−17.62168639091104384225850244016, −17.10084375870744496675948120153, −16.36592037300909954977019701931, −15.70077268152690786234085537479, −14.84590629283205855088175006227, −14.08784873069107229879710933788, −13.698820577743609430852340355990, −12.96839095396024008717615847374, −12.45837473571945063885731925063, −11.467902888401352713371211774107, −10.70904967366182765836593114101, −10.280351584800320687885994490506, −9.54234310236462538419525456862, −8.71470375482543976605908916170, −8.23777944016969965791935758411, −7.253527104628447447628656883308, −6.5976139538521679725691473015, −5.937355193840405994650907313563, −5.122763852930450487856234607720, −4.33219374471814320257077405550, −3.85818184866509054231865486079, −2.61710764246879918128911269412, −1.89901923996613291062210277347, −1.2288890104463694645169828818, −0.00985518235282826471684144197,
1.64356581109101492391957852981, 2.08884524236399026050741960681, 2.76389435877372946901270379254, 3.72485819312184541945225896214, 4.68546697692243789984083166172, 5.36957116098899686371040334059, 6.106711993616884833109557578105, 6.55360440159006067071115806155, 7.47383785574365012418021404117, 8.36385094788029767693783209302, 8.96316195508786041225012120119, 9.58416716916759345925086672813, 10.31025877931075690012777692134, 11.03407316209953060236523243455, 11.63082558565894872936455357398, 12.44623290537134028411988418237, 13.13435707582195102963282351848, 13.68608999006492969120095455143, 14.65508573170263033947232757357, 14.81944894597946642780782200521, 15.73145431474817739578656053453, 16.393483353040837619374330072764, 17.2199489345658056688871468153, 17.8963287917353812715835056720, 18.19273637085845577676817906596