| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 290.·5-s − 216·6-s + 419.·7-s − 512·8-s + 729·9-s + 2.32e3·10-s + 119.·11-s + 1.72e3·12-s − 4.14e3·13-s − 3.35e3·14-s − 7.84e3·15-s + 4.09e3·16-s + 2.61e4·17-s − 5.83e3·18-s − 1.53e4·19-s − 1.86e4·20-s + 1.13e4·21-s − 952.·22-s − 7.25e4·23-s − 1.38e4·24-s + 6.37e3·25-s + 3.31e4·26-s + 1.96e4·27-s + 2.68e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.04·5-s − 0.408·6-s + 0.462·7-s − 0.353·8-s + 0.333·9-s + 0.735·10-s + 0.0269·11-s + 0.288·12-s − 0.523·13-s − 0.326·14-s − 0.600·15-s + 0.250·16-s + 1.29·17-s − 0.235·18-s − 0.514·19-s − 0.520·20-s + 0.266·21-s − 0.0190·22-s − 1.24·23-s − 0.204·24-s + 0.0816·25-s + 0.370·26-s + 0.192·27-s + 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 5 | \( 1 + 290.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 419.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 119.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.14e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.61e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.53e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.63e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.79e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.95e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.48e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.62e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.98e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 6.98e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.87e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.26e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.26e7T + 8.07e13T^{2} \) |
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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
−9.841935399853054312889744678523, −8.666382456523756310882277465287, −7.938053071163317289422375980100, −7.47259927316547479299381067473, −6.16985924040066941109940145104, −4.68509259028669757062422942284, −3.64180142749328335050264357604, −2.50085066162538318008757041273, −1.22368504628631851236296697738, 0,
1.22368504628631851236296697738, 2.50085066162538318008757041273, 3.64180142749328335050264357604, 4.68509259028669757062422942284, 6.16985924040066941109940145104, 7.47259927316547479299381067473, 7.938053071163317289422375980100, 8.666382456523756310882277465287, 9.841935399853054312889744678523
