| L(s) = 1 | + 2.23·5-s + 2.38·7-s + 2.23·11-s − 6.23·13-s − 1.85·17-s − 3.09·19-s − 4.61·23-s − 6.38·29-s + 10.5·31-s + 5.32·35-s − 0.145·37-s − 8.09·41-s + 8.70·43-s − 10.8·47-s − 1.32·49-s − 6.23·53-s + 5.00·55-s − 59-s − 3.14·61-s − 13.9·65-s − 10.7·67-s − 7.94·71-s + 0.854·73-s + 5.32·77-s + 3·79-s − 1.61·83-s − 4.14·85-s + ⋯ |
| L(s) = 1 | + 0.999·5-s + 0.900·7-s + 0.674·11-s − 1.72·13-s − 0.449·17-s − 0.708·19-s − 0.962·23-s − 1.18·29-s + 1.89·31-s + 0.900·35-s − 0.0239·37-s − 1.26·41-s + 1.32·43-s − 1.58·47-s − 0.189·49-s − 0.856·53-s + 0.674·55-s − 0.130·59-s − 0.402·61-s − 1.72·65-s − 1.30·67-s − 0.942·71-s + 0.0999·73-s + 0.606·77-s + 0.337·79-s − 0.177·83-s − 0.449·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 + 6.38T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.145T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 61 | \( 1 + 3.14T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 - 0.854T + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 3T + 97T^{2} \) |
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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
−7.59442353917188211030365393185, −6.54894844594952082407828683153, −6.21440393329445255174025412508, −5.25163930211797405777827127582, −4.72205384117052435349873991676, −4.08007924038048788011554572519, −2.85120063986225930155029347099, −2.06136522244420388030348005983, −1.56177360104495315315923687195, 0,
1.56177360104495315315923687195, 2.06136522244420388030348005983, 2.85120063986225930155029347099, 4.08007924038048788011554572519, 4.72205384117052435349873991676, 5.25163930211797405777827127582, 6.21440393329445255174025412508, 6.54894844594952082407828683153, 7.59442353917188211030365393185
