| L(s) = 1 | + 0.841i·5-s − 2.64·7-s − 3.91i·11-s − 0.645i·13-s − 5.59·17-s + 4.29i·19-s − 8.66·23-s + 4.29·25-s − 7.82i·29-s − 2·31-s − 2.22i·35-s − 6.64i·37-s − 6.13·41-s − 7.29i·43-s − 2.52·47-s + ⋯ |
| L(s) = 1 | + 0.376i·5-s − 0.999·7-s − 1.17i·11-s − 0.179i·13-s − 1.35·17-s + 0.984i·19-s − 1.80·23-s + 0.858·25-s − 1.45i·29-s − 0.359·31-s − 0.376i·35-s − 1.09i·37-s − 0.958·41-s − 1.11i·43-s − 0.368·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100824 - 0.341994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100824 - 0.341994i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.841iT - 5T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 + 3.91iT - 11T^{2} \) |
| 13 | \( 1 + 0.645iT - 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 - 4.29iT - 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 + 7.82iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6.64iT - 37T^{2} \) |
| 41 | \( 1 + 6.13T + 41T^{2} \) |
| 43 | \( 1 + 7.29iT - 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 + 7.82iT - 53T^{2} \) |
| 59 | \( 1 - 7.27iT - 59T^{2} \) |
| 61 | \( 1 - 13.9iT - 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 + 9.50iT - 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−9.976820335026263913639607260268, −8.923484467563576007660393117622, −8.244467651297032666547414659259, −7.15494846073948520697513161856, −6.22950956177418547352795145633, −5.74005819420315550583154144238, −4.16805454259222459642881691746, −3.36768107669615223888147751225, −2.21520294651378898806180343797, −0.15972764853230071100038529900,
1.84776583630653514169988731410, 3.08380335417315169784650282551, 4.34368708133810703996311186463, 5.04754488532750580852491107368, 6.49933696057365717267769173395, 6.81663302968093276727432095835, 8.016137377282958566707202773531, 9.015364710748495936632730991220, 9.572205160207407447070474774701, 10.38766997471153093009247752971
