| L(s) = 1 | − 2-s + 4-s − 0.445·5-s − 8-s + 9-s + 0.445·10-s + 1.24·13-s + 16-s + 1.80·17-s − 18-s − 0.445·20-s − 0.801·25-s − 1.24·26-s − 32-s − 1.80·34-s + 36-s − 1.24·37-s + 0.445·40-s − 1.24·41-s − 0.445·45-s + 49-s + 0.801·50-s + 1.24·52-s − 1.80·53-s + 1.80·61-s + 64-s − 0.554·65-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 0.445·5-s − 8-s + 9-s + 0.445·10-s + 1.24·13-s + 16-s + 1.80·17-s − 18-s − 0.445·20-s − 0.801·25-s − 1.24·26-s − 32-s − 1.80·34-s + 36-s − 1.24·37-s + 0.445·40-s − 1.24·41-s − 0.445·45-s + 49-s + 0.801·50-s + 1.24·52-s − 1.80·53-s + 1.80·61-s + 64-s − 0.554·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9010881936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9010881936\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.24T + T^{2} \) |
| 17 | \( 1 - 1.80T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.24T + T^{2} \) |
| 41 | \( 1 + 1.24T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.80T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.445T + T^{2} \) |
| 97 | \( 1 + 1.24T + T^{2} \) |
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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
−8.659601124009634377879746825502, −8.101495026469937817480184029436, −7.47709001309989253339558478685, −6.78578388401659292837254197821, −5.98559493231980408096428427340, −5.14352511238117401696326059677, −3.78078859792851565068141242302, −3.36421363108133430410468303240, −1.89686527629691327744203762069, −1.04586633872361525407601006603,
1.04586633872361525407601006603, 1.89686527629691327744203762069, 3.36421363108133430410468303240, 3.78078859792851565068141242302, 5.14352511238117401696326059677, 5.98559493231980408096428427340, 6.78578388401659292837254197821, 7.47709001309989253339558478685, 8.101495026469937817480184029436, 8.659601124009634377879746825502
