| L(s) = 1 | − 2.15·2-s + 2.65·4-s + 2.23·5-s − 3.60·7-s − 1.41·8-s − 4.82·10-s + 1.50·11-s + 0.00459·13-s + 7.77·14-s − 2.25·16-s − 1.11·17-s + 2.49·19-s + 5.93·20-s − 3.24·22-s − 1.72·23-s − 0.00724·25-s − 0.00991·26-s − 9.57·28-s + 0.572·29-s + 3.25·31-s + 7.70·32-s + 2.41·34-s − 8.05·35-s − 6.13·37-s − 5.39·38-s − 3.16·40-s + 5.89·41-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 1.32·4-s + 0.999·5-s − 1.36·7-s − 0.500·8-s − 1.52·10-s + 0.453·11-s + 0.00127·13-s + 2.07·14-s − 0.564·16-s − 0.271·17-s + 0.573·19-s + 1.32·20-s − 0.691·22-s − 0.359·23-s − 0.00144·25-s − 0.00194·26-s − 1.80·28-s + 0.106·29-s + 0.584·31-s + 1.36·32-s + 0.414·34-s − 1.36·35-s − 1.00·37-s − 0.874·38-s − 0.500·40-s + 0.920·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 - 0.00459T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 - 0.572T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 7.82T + 53T^{2} \) |
| 59 | \( 1 + 0.253T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 1.99T + 79T^{2} \) |
| 83 | \( 1 + 6.91T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.230888569133009719225264537655, −7.21451246205720319376934993481, −6.69081045851740850383640211072, −6.13040899463083916827989557723, −5.27139492576520482684408974651, −4.03729654711119243954426135019, −3.00580031855116554589193391084, −2.14524709198601610545050107284, −1.20801783555688648875380750447, 0,
1.20801783555688648875380750447, 2.14524709198601610545050107284, 3.00580031855116554589193391084, 4.03729654711119243954426135019, 5.27139492576520482684408974651, 6.13040899463083916827989557723, 6.69081045851740850383640211072, 7.21451246205720319376934993481, 8.230888569133009719225264537655
