L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.939 + 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.0603 − 0.342i)17-s + (0.939 + 0.342i)18-s + (0.766 − 0.642i)19-s + (0.766 + 0.642i)20-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.939 + 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.0603 − 0.342i)17-s + (0.939 + 0.342i)18-s + (0.766 − 0.642i)19-s + (0.766 + 0.642i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9129660540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9129660540\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.488977128787791432325612075696, −8.895385459439858727099972480394, −8.117784081769897292043711190746, −7.60233674339251996231711554784, −6.84809041572377693254632552005, −5.98891700824936398553731938705, −4.70114670213310157930325854952, −3.99300200869724195453419822092, −2.73180593234324893404649505224, −0.867820229552478453820520797292,
1.75714343176139494163699574679, 3.14978638577712685956646717045, 3.58796208388354551168220102217, 4.51400287804260659965883315041, 5.46231795902755239513262213844, 7.13094320259581062469764166932, 7.993694831255911667349897169450, 8.456146527395147428573826921584, 9.439831292688013819382044545504, 10.09868774481851663952603559801