L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.142 − 0.989i)3-s + (−0.142 + 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.841 + 0.540i)6-s + (0.841 − 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)10-s + (0.959 + 0.281i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (1.65 − 0.755i)17-s + (0.415 + 0.909i)18-s + (1.37 + 0.627i)19-s + (0.142 + 0.989i)20-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.142 − 0.989i)3-s + (−0.142 + 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.841 + 0.540i)6-s + (0.841 − 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)10-s + (0.959 + 0.281i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (1.65 − 0.755i)17-s + (0.415 + 0.909i)18-s + (1.37 + 0.627i)19-s + (0.142 + 0.989i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.142013216\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.142013216\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-0.841 - 0.540i)T \) |
good | 7 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (-1.65 + 0.755i)T + (0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 0.627i)T + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (1.49 - 0.215i)T + (0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - 1.08iT - T^{2} \) |
| 53 | \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.557 + 1.89i)T + (-0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 - 0.540i)T^{2} \) |
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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
−8.994580202080264977143372973859, −7.88435866167206753347694226961, −7.57464777023880488142293148231, −6.69652068727261095983490329389, −5.64454829770204505739231365794, −5.06307582179603976824180796502, −3.41326219846391613915295040935, −2.91588692908030763920200763355, −1.70187520718880059204330783762, −1.10245396466966919438288208418,
1.35249137706804468664053459871, 2.68221531761419465860553469974, 3.64220934666856922841544830401, 4.93026487947649144467539933646, 5.46899618718725283061174707311, 6.01372257294055893700213794027, 7.05974137464429443038497412505, 7.73309494624639151902362477853, 8.703737804169050293516885754139, 9.234301243675967860623976019148