| L(s) = 1 | − 52.4·3-s − 199.·5-s + 559.·9-s + 1.21e3·11-s − 6.44e3·13-s + 1.04e4·15-s − 2.67e4·17-s − 1.49e4·19-s − 7.35e4·23-s − 3.84e4·25-s + 8.52e4·27-s − 1.06e5·29-s − 6.68e4·31-s − 6.38e4·33-s − 4.79e5·37-s + 3.37e5·39-s + 6.44e5·41-s + 1.45e5·43-s − 1.11e5·45-s − 1.01e6·47-s + 1.40e6·51-s + 3.42e4·53-s − 2.42e5·55-s + 7.81e5·57-s + 4.43e5·59-s + 1.14e6·61-s + 1.28e6·65-s + ⋯ |
| L(s) = 1 | − 1.12·3-s − 0.712·5-s + 0.255·9-s + 0.276·11-s − 0.814·13-s + 0.798·15-s − 1.32·17-s − 0.498·19-s − 1.25·23-s − 0.492·25-s + 0.834·27-s − 0.810·29-s − 0.402·31-s − 0.309·33-s − 1.55·37-s + 0.912·39-s + 1.46·41-s + 0.278·43-s − 0.182·45-s − 1.42·47-s + 1.48·51-s + 0.0315·53-s − 0.196·55-s + 0.558·57-s + 0.281·59-s + 0.643·61-s + 0.580·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2853225128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2853225128\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 52.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 199.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 6.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.42e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.72e7T + 8.07e13T^{2} \) |
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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
−11.32616347898286468158999223711, −10.52332356476606301079289914380, −9.291923717008926983276451698270, −8.088211479974850341904686639693, −6.96973106701255457967705731676, −6.01724464749207685620614276130, −4.87174003378460349747531716126, −3.85899188725691580977576345406, −2.09094605178960417862871090500, −0.28741891428630805851299827376,
0.28741891428630805851299827376, 2.09094605178960417862871090500, 3.85899188725691580977576345406, 4.87174003378460349747531716126, 6.01724464749207685620614276130, 6.96973106701255457967705731676, 8.088211479974850341904686639693, 9.291923717008926983276451698270, 10.52332356476606301079289914380, 11.32616347898286468158999223711
