| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 320.·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s − 2.56e3·10-s − 1.11e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s + 8.65e3·15-s + 4.09e3·16-s + 3.82e3·17-s − 5.83e3·18-s − 4.53e4·19-s + 2.05e4·20-s + 9.26e3·21-s + 8.88e3·22-s − 5.25e3·23-s − 1.38e4·24-s + 2.46e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.811·10-s − 0.251·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.662·15-s + 0.250·16-s + 0.188·17-s − 0.235·18-s − 1.51·19-s + 0.573·20-s + 0.218·21-s + 0.177·22-s − 0.0900·23-s − 0.204·24-s + 0.315·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 5 | \( 1 - 320.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.11e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.82e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.53e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.25e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.51e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.23e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.27e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.84e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.18e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.74e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.94e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.62e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.88e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.69e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.69e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.85e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.65e7T + 8.07e13T^{2} \) |
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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
−9.180538452875186411788438940402, −8.493001304567824689691687824590, −7.65002192479866654601287951537, −6.55083243961363390693763173286, −5.77390134278348588893507215249, −4.55404309607707753848088365467, −3.17302582209710287068071380597, −2.05972447667566170114915713611, −1.51971075104965164580197530872, 0,
1.51971075104965164580197530872, 2.05972447667566170114915713611, 3.17302582209710287068071380597, 4.55404309607707753848088365467, 5.77390134278348588893507215249, 6.55083243961363390693763173286, 7.65002192479866654601287951537, 8.493001304567824689691687824590, 9.180538452875186411788438940402
