L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 7-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s + 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)23-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 7-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s + 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5695104506\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5695104506\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−11.80071508954180853926845378276, −11.20615371565957059761013023842, −9.860082458716909779214524538077, −8.973991931372176462136622755714, −8.292953158098930586523783564286, −7.38314043813070187558221167778, −5.53551928462064910010219100853, −4.38038338526091381188467776575, −3.23446085663219780964219393327, −1.29490937665423030484310327694,
2.30247443887687932688881450063, 4.45199213217626417055744200907, 5.26808382279490391258148059126, 6.79736537383575133286069677998, 7.45877146856009408561287802246, 8.244697824609083528068906561575, 9.368086750882871791009580674703, 10.48062927226413804045761639682, 11.13932298076329041515698727402, 12.13830326851865339363536546666