| L(s) = 1 | − 17.4·2-s − 27·3-s + 175.·4-s + 74.3·5-s + 469.·6-s − 818.·8-s + 729·9-s − 1.29e3·10-s − 1.95e3·11-s − 4.72e3·12-s − 3.40e3·13-s − 2.00e3·15-s − 8.15e3·16-s − 2.23e4·17-s − 1.26e4·18-s + 4.98e4·19-s + 1.30e4·20-s + 3.40e4·22-s − 1.19e4·23-s + 2.20e4·24-s − 7.25e4·25-s + 5.93e4·26-s − 1.96e4·27-s + 1.62e5·29-s + 3.49e4·30-s + 3.16e4·31-s + 2.46e5·32-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.36·4-s + 0.266·5-s + 0.888·6-s − 0.564·8-s + 0.333·9-s − 0.409·10-s − 0.443·11-s − 0.789·12-s − 0.430·13-s − 0.153·15-s − 0.497·16-s − 1.10·17-s − 0.512·18-s + 1.66·19-s + 0.363·20-s + 0.682·22-s − 0.205·23-s + 0.326·24-s − 0.929·25-s + 0.661·26-s − 0.192·27-s + 1.24·29-s + 0.236·30-s + 0.190·31-s + 1.33·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 17.4T + 128T^{2} \) |
| 5 | \( 1 - 74.3T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.95e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.40e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.23e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.98e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.62e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.16e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.74e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.52e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.54e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.20e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.16e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.18e6T + 8.07e13T^{2} \) |
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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
−11.01559004356365751867951242694, −9.964931629998126823190810451065, −9.384624017011376911134177352764, −8.094890075015120784702782427819, −7.23807394750837590383750204236, −6.06946905178004382624872361925, −4.64872743145174874538169670379, −2.52789568900655027190940835536, −1.15774658953126606397052722072, 0,
1.15774658953126606397052722072, 2.52789568900655027190940835536, 4.64872743145174874538169670379, 6.06946905178004382624872361925, 7.23807394750837590383750204236, 8.094890075015120784702782427819, 9.384624017011376911134177352764, 9.964931629998126823190810451065, 11.01559004356365751867951242694
