| L(s) = 1 | + 1.07·3-s − 3.54·5-s + 3.82·7-s − 1.85·9-s + 4.38·11-s + 5.21·13-s − 3.79·15-s − 0.517·17-s + 3.99·19-s + 4.09·21-s + 7.57·25-s − 5.19·27-s − 9.23·29-s − 3.51·31-s + 4.69·33-s − 13.5·35-s + 3.44·37-s + 5.57·39-s + 10.8·41-s + 0.391·43-s + 6.57·45-s + 1.50·47-s + 7.66·49-s − 0.553·51-s + 0.746·53-s − 15.5·55-s + 4.27·57-s + ⋯ |
| L(s) = 1 | + 0.618·3-s − 1.58·5-s + 1.44·7-s − 0.617·9-s + 1.32·11-s + 1.44·13-s − 0.980·15-s − 0.125·17-s + 0.916·19-s + 0.894·21-s + 1.51·25-s − 1.00·27-s − 1.71·29-s − 0.631·31-s + 0.817·33-s − 2.29·35-s + 0.565·37-s + 0.893·39-s + 1.68·41-s + 0.0596·43-s + 0.979·45-s + 0.220·47-s + 1.09·49-s − 0.0775·51-s + 0.102·53-s − 2.09·55-s + 0.566·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.337396384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.337396384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 - 5.21T + 13T^{2} \) |
| 17 | \( 1 + 0.517T + 17T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 0.391T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 - 0.746T + 53T^{2} \) |
| 59 | \( 1 + 5.69T + 59T^{2} \) |
| 61 | \( 1 - 5.94T + 61T^{2} \) |
| 67 | \( 1 - 2.72T + 67T^{2} \) |
| 71 | \( 1 + 3.85T + 71T^{2} \) |
| 73 | \( 1 + 5.78T + 73T^{2} \) |
| 79 | \( 1 - 2.75T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + 1.93T + 97T^{2} \) |
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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
−8.412715147283185673944961865411, −7.65530332145278229690717808819, −7.40467365594429912174895814265, −6.17885242741637947030258590472, −5.39590870286058094309655013172, −4.28513628591512439368338337895, −3.87012020070567924126725631352, −3.20930290084213196909223600685, −1.85933791353537352931584059427, −0.892868954002520655119999180288,
0.892868954002520655119999180288, 1.85933791353537352931584059427, 3.20930290084213196909223600685, 3.87012020070567924126725631352, 4.28513628591512439368338337895, 5.39590870286058094309655013172, 6.17885242741637947030258590472, 7.40467365594429912174895814265, 7.65530332145278229690717808819, 8.412715147283185673944961865411
