| L(s) = 1 | + 2.43·3-s + 5-s + 7-s + 2.94·9-s − 1.23·11-s + 13-s + 2.43·15-s + 0.931·17-s + 7.18·19-s + 2.43·21-s − 2.65·23-s + 25-s − 0.133·27-s + 2.66·29-s + 7.04·31-s − 3.01·33-s + 35-s − 0.797·37-s + 2.43·39-s − 2.52·41-s + 3.64·43-s + 2.94·45-s − 1.05·47-s + 49-s + 2.27·51-s + 5.89·53-s − 1.23·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 0.447·5-s + 0.377·7-s + 0.981·9-s − 0.372·11-s + 0.277·13-s + 0.629·15-s + 0.225·17-s + 1.64·19-s + 0.532·21-s − 0.553·23-s + 0.200·25-s − 0.0257·27-s + 0.494·29-s + 1.26·31-s − 0.524·33-s + 0.169·35-s − 0.131·37-s + 0.390·39-s − 0.394·41-s + 0.555·43-s + 0.439·45-s − 0.153·47-s + 0.142·49-s + 0.317·51-s + 0.809·53-s − 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.327146743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.327146743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 17 | \( 1 - 0.931T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 0.797T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 1.05T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 + 0.349T + 67T^{2} \) |
| 71 | \( 1 + 0.249T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 - 6.75T + 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 + 6.64T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−7.976819917499115594469783080232, −7.50227004995874400016338903035, −6.64302539289701874289895547466, −5.76312921480347872451517608278, −5.06264526046518132798434512311, −4.20649880416885501164822508446, −3.31779108938254101473051819465, −2.78379518849021242161502258142, −1.96145810285373161464107585743, −1.03727949208384982708194364556,
1.03727949208384982708194364556, 1.96145810285373161464107585743, 2.78379518849021242161502258142, 3.31779108938254101473051819465, 4.20649880416885501164822508446, 5.06264526046518132798434512311, 5.76312921480347872451517608278, 6.64302539289701874289895547466, 7.50227004995874400016338903035, 7.976819917499115594469783080232
