| L(s) = 1 | + 303.·5-s − 615.·7-s − 2.45e3·11-s − 284.·13-s + 6.23e3·17-s + 4.25e4·19-s − 2.35e4·23-s + 1.39e4·25-s − 1.96e5·29-s − 1.60e5·31-s − 1.86e5·35-s + 3.18e5·37-s − 2.86e5·41-s − 1.57e5·43-s + 3.43e5·47-s − 4.45e5·49-s − 2.53e5·53-s − 7.46e5·55-s + 1.67e6·59-s − 1.07e6·61-s − 8.63e4·65-s + 1.58e6·67-s + 1.72e6·71-s − 4.35e6·73-s + 1.51e6·77-s + 4.17e6·79-s − 2.84e6·83-s + ⋯ |
| L(s) = 1 | + 1.08·5-s − 0.677·7-s − 0.556·11-s − 0.0359·13-s + 0.307·17-s + 1.42·19-s − 0.403·23-s + 0.178·25-s − 1.49·29-s − 0.966·31-s − 0.735·35-s + 1.03·37-s − 0.649·41-s − 0.301·43-s + 0.481·47-s − 0.540·49-s − 0.233·53-s − 0.604·55-s + 1.06·59-s − 0.606·61-s − 0.0389·65-s + 0.642·67-s + 0.572·71-s − 1.31·73-s + 0.377·77-s + 0.951·79-s − 0.545·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 303.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 615.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 284.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.23e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.25e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.96e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.60e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.86e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.53e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.67e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.07e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.58e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.17e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.84e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.30e7T + 8.07e13T^{2} \) |
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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
−9.687034724832136295021479973052, −9.425586019723841194251050735629, −7.985154717112168832139738619044, −7.00539404248681395581140491347, −5.85885625989694821215590399018, −5.28284129137038595256133929337, −3.68052493033001903815020547221, −2.58566011455571719824600531351, −1.44976546787586841530459266000, 0,
1.44976546787586841530459266000, 2.58566011455571719824600531351, 3.68052493033001903815020547221, 5.28284129137038595256133929337, 5.85885625989694821215590399018, 7.00539404248681395581140491347, 7.985154717112168832139738619044, 9.425586019723841194251050735629, 9.687034724832136295021479973052
