| L(s) = 1 | + 1.48·3-s + 3.18·5-s − 3.31·7-s − 0.793·9-s − 0.867·11-s − 2.69·13-s + 4.73·15-s + 2.03·17-s − 8.43·19-s − 4.91·21-s + 6.66·23-s + 5.14·25-s − 5.63·27-s − 7.02·29-s − 4.72·31-s − 1.28·33-s − 10.5·35-s − 1.00·37-s − 4.00·39-s + 11.7·41-s − 1.30·43-s − 2.52·45-s − 5.57·47-s + 3.97·49-s + 3.02·51-s − 5.95·53-s − 2.76·55-s + ⋯ |
| L(s) = 1 | + 0.857·3-s + 1.42·5-s − 1.25·7-s − 0.264·9-s − 0.261·11-s − 0.747·13-s + 1.22·15-s + 0.493·17-s − 1.93·19-s − 1.07·21-s + 1.38·23-s + 1.02·25-s − 1.08·27-s − 1.30·29-s − 0.848·31-s − 0.224·33-s − 1.78·35-s − 0.165·37-s − 0.640·39-s + 1.82·41-s − 0.199·43-s − 0.376·45-s − 0.813·47-s + 0.567·49-s + 0.423·51-s − 0.817·53-s − 0.372·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 + 0.867T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 2.03T + 17T^{2} \) |
| 19 | \( 1 + 8.43T + 19T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 + 1.00T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 - 0.698T + 73T^{2} \) |
| 79 | \( 1 - 3.24T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 6.06T + 89T^{2} \) |
| 97 | \( 1 + 7.76T + 97T^{2} \) |
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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
−8.132990615005969296661931998879, −7.23112832077786520343468118403, −6.51611127601154433720487330687, −5.85351250745129015785207560926, −5.23374110035147950287181168966, −4.05867125238433398732176349523, −3.07952971693879330318221056637, −2.54168558773085624465513847514, −1.76941416862055347408073050612, 0,
1.76941416862055347408073050612, 2.54168558773085624465513847514, 3.07952971693879330318221056637, 4.05867125238433398732176349523, 5.23374110035147950287181168966, 5.85351250745129015785207560926, 6.51611127601154433720487330687, 7.23112832077786520343468118403, 8.132990615005969296661931998879
