L(s) = 1 | + 2.23·5-s + 7-s − 2.85·11-s − 1.76·13-s − 0.854·17-s + 19-s − 6.23·23-s + 4.38·29-s − 2.38·31-s + 2.23·35-s + 1.47·37-s + 10.5·41-s − 6.94·43-s − 7·47-s + 49-s + 1.85·53-s − 6.38·55-s + 10.7·59-s − 12.7·61-s − 3.94·65-s − 5.61·67-s − 3.94·71-s − 0.145·73-s − 2.85·77-s − 4.47·79-s + 11.3·83-s − 1.90·85-s + ⋯ |
L(s) = 1 | + 0.999·5-s + 0.377·7-s − 0.860·11-s − 0.489·13-s − 0.207·17-s + 0.229·19-s − 1.30·23-s + 0.813·29-s − 0.427·31-s + 0.377·35-s + 0.242·37-s + 1.64·41-s − 1.05·43-s − 1.02·47-s + 0.142·49-s + 0.254·53-s − 0.860·55-s + 1.39·59-s − 1.62·61-s − 0.489·65-s − 0.686·67-s − 0.468·71-s − 0.0170·73-s − 0.325·77-s − 0.503·79-s + 1.24·83-s − 0.207·85-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 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 0.854T + 17T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 + 0.145T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 0.763T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−7.49518463384154262959322128698, −6.52492701833764503946545740076, −5.96909335183284681638111578499, −5.31188770468733093912461425437, −4.72229861920770328497778172301, −3.87566691652684810545411004986, −2.77597498803856012760135702109, −2.22674714033674305834178525694, −1.38946216191953009008290482688, 0,
1.38946216191953009008290482688, 2.22674714033674305834178525694, 2.77597498803856012760135702109, 3.87566691652684810545411004986, 4.72229861920770328497778172301, 5.31188770468733093912461425437, 5.96909335183284681638111578499, 6.52492701833764503946545740076, 7.49518463384154262959322128698