| L(s) = 1 | + 8.56·5-s + 1.27e3·7-s + 3.79e3·11-s − 5.18e3·13-s − 3.08e4·17-s − 3.12e4·19-s − 2.66e4·23-s − 7.80e4·25-s + 1.75e5·29-s − 1.17e5·31-s + 1.08e4·35-s + 9.33e4·37-s + 2.41e5·41-s + 1.81e5·43-s − 1.19e6·47-s + 7.95e5·49-s + 4.57e5·53-s + 3.24e4·55-s − 2.41e5·59-s + 9.67e5·61-s − 4.43e4·65-s − 2.46e6·67-s + 7.15e5·71-s − 1.12e6·73-s + 4.82e6·77-s − 7.89e6·79-s + 2.58e6·83-s + ⋯ |
| L(s) = 1 | + 0.0306·5-s + 1.40·7-s + 0.859·11-s − 0.653·13-s − 1.52·17-s − 1.04·19-s − 0.455·23-s − 0.999·25-s + 1.33·29-s − 0.706·31-s + 0.0429·35-s + 0.303·37-s + 0.548·41-s + 0.347·43-s − 1.67·47-s + 0.965·49-s + 0.422·53-s + 0.0263·55-s − 0.153·59-s + 0.545·61-s − 0.0200·65-s − 1.00·67-s + 0.237·71-s − 0.338·73-s + 1.20·77-s − 1.80·79-s + 0.496·83-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 | \( 1 - 8.56T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.27e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.79e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.08e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.66e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.75e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.17e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 9.33e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.81e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.19e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.57e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.41e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.67e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 7.15e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.89e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.58e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.25e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.69e6T + 8.07e13T^{2} \) |
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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
−9.963231704701336889798680031020, −8.850621014075768810464610618176, −8.164596815719529623819926979298, −7.06928341056278331503039881489, −6.06679231533844044187836819074, −4.73902386021434899162744823465, −4.13684931371878675010676464469, −2.36956683465844057940237776856, −1.51733974237784045653681015420, 0,
1.51733974237784045653681015420, 2.36956683465844057940237776856, 4.13684931371878675010676464469, 4.73902386021434899162744823465, 6.06679231533844044187836819074, 7.06928341056278331503039881489, 8.164596815719529623819926979298, 8.850621014075768810464610618176, 9.963231704701336889798680031020
