| L(s) = 1 | + 1.04·2-s − 126.·4-s + 45.6·5-s − 399.·7-s − 266.·8-s + 47.7·10-s + 7.63e3·11-s − 1.07e4·13-s − 417.·14-s + 1.59e4·16-s − 1.62e4·17-s − 1.16e3·19-s − 5.79e3·20-s + 7.98e3·22-s + 4.86e4·23-s − 7.60e4·25-s − 1.13e4·26-s + 5.06e4·28-s − 1.18e4·29-s − 6.29e4·31-s + 5.08e4·32-s − 1.70e4·34-s − 1.82e4·35-s + 7.93e4·37-s − 1.22e3·38-s − 1.21e4·40-s + 6.34e5·41-s + ⋯ |
| L(s) = 1 | + 0.0925·2-s − 0.991·4-s + 0.163·5-s − 0.439·7-s − 0.184·8-s + 0.0151·10-s + 1.72·11-s − 1.36·13-s − 0.0406·14-s + 0.974·16-s − 0.803·17-s − 0.0389·19-s − 0.161·20-s + 0.159·22-s + 0.833·23-s − 0.973·25-s − 0.126·26-s + 0.435·28-s − 0.0902·29-s − 0.379·31-s + 0.274·32-s − 0.0743·34-s − 0.0717·35-s + 0.257·37-s − 0.00360·38-s − 0.0300·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 1.04T + 128T^{2} \) |
| 5 | \( 1 - 45.6T + 7.81e4T^{2} \) |
| 7 | \( 1 + 399.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.07e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.62e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.16e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.18e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 6.29e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 7.93e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.92e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.82e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.82e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.18e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.62e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 8.35e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.93e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.75e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.61e6T + 8.07e13T^{2} \) |
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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
−9.484815782090185969434944993079, −8.610688079408259901934737311886, −7.40298141147057829792955858082, −6.50974713035359832295729775400, −5.49652936581499257572582462180, −4.42652192904734999356984981987, −3.76571819221071710767441142312, −2.43239886254641481750617467239, −1.06204956081792448616983201514, 0,
1.06204956081792448616983201514, 2.43239886254641481750617467239, 3.76571819221071710767441142312, 4.42652192904734999356984981987, 5.49652936581499257572582462180, 6.50974713035359832295729775400, 7.40298141147057829792955858082, 8.610688079408259901934737311886, 9.484815782090185969434944993079
