| L(s) = 1 | + 9·3-s − 53.5·5-s − 8.43·7-s + 81·9-s + 245.·11-s + 412.·13-s − 481.·15-s − 1.29e3·17-s − 379.·19-s − 75.9·21-s + 529·23-s − 258.·25-s + 729·27-s + 4.13e3·29-s − 4.92e3·31-s + 2.21e3·33-s + 451.·35-s + 1.28e3·37-s + 3.71e3·39-s + 5.05e3·41-s + 1.85e4·43-s − 4.33e3·45-s − 5.90e3·47-s − 1.67e4·49-s − 1.16e4·51-s + 3.57e4·53-s − 1.31e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.957·5-s − 0.0650·7-s + 0.333·9-s + 0.612·11-s + 0.676·13-s − 0.552·15-s − 1.08·17-s − 0.241·19-s − 0.0375·21-s + 0.208·23-s − 0.0826·25-s + 0.192·27-s + 0.913·29-s − 0.919·31-s + 0.353·33-s + 0.0623·35-s + 0.153·37-s + 0.390·39-s + 0.469·41-s + 1.53·43-s − 0.319·45-s − 0.389·47-s − 0.995·49-s − 0.625·51-s + 1.74·53-s − 0.586·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 + 53.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 8.43T + 1.68e4T^{2} \) |
| 11 | \( 1 - 245.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 412.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.29e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 379.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.24e5T + 8.58e9T^{2} \) |
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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
−9.284579963877305769564122983442, −8.717399967973897124857892876642, −7.83283981422645974173604060782, −6.98658184225311417870610104098, −6.01780016520397557384145867496, −4.47778306611212419287302548625, −3.87130013091368192053409492735, −2.75905525689061727750314694702, −1.39045609954342812871232121947, 0,
1.39045609954342812871232121947, 2.75905525689061727750314694702, 3.87130013091368192053409492735, 4.47778306611212419287302548625, 6.01780016520397557384145867496, 6.98658184225311417870610104098, 7.83283981422645974173604060782, 8.717399967973897124857892876642, 9.284579963877305769564122983442
