| L(s) = 1 | + 3.23·5-s − 7-s − 0.763·11-s − 5.23·13-s + 2.76·17-s + 19-s + 1.23·23-s + 5.47·25-s − 0.472·29-s − 8.47·31-s − 3.23·35-s − 8.94·37-s + 2·41-s − 4·43-s + 2.47·47-s + 49-s + 8.47·53-s − 2.47·55-s − 8·59-s − 0.472·61-s − 16.9·65-s + 5.23·67-s − 10.4·71-s − 4.47·73-s + 0.763·77-s + 2.29·79-s + 2·83-s + ⋯ |
| L(s) = 1 | + 1.44·5-s − 0.377·7-s − 0.230·11-s − 1.45·13-s + 0.670·17-s + 0.229·19-s + 0.257·23-s + 1.09·25-s − 0.0876·29-s − 1.52·31-s − 0.546·35-s − 1.47·37-s + 0.312·41-s − 0.609·43-s + 0.360·47-s + 0.142·49-s + 1.16·53-s − 0.333·55-s − 1.04·59-s − 0.0604·61-s − 2.10·65-s + 0.639·67-s − 1.24·71-s − 0.523·73-s + 0.0870·77-s + 0.257·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−7.21485225397475412147114786577, −6.74501910984697976146392828346, −5.79296642442226513235641897268, −5.41620808211978932750062680226, −4.82872440384361548174008013631, −3.71236298188686259298093930437, −2.86150048264250425115867151696, −2.19628579670689065891934734060, −1.40770261180097262244541817008, 0,
1.40770261180097262244541817008, 2.19628579670689065891934734060, 2.86150048264250425115867151696, 3.71236298188686259298093930437, 4.82872440384361548174008013631, 5.41620808211978932750062680226, 5.79296642442226513235641897268, 6.74501910984697976146392828346, 7.21485225397475412147114786577
