L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 56.7·5-s + 216·6-s + 343·7-s + 512·8-s + 729·9-s − 453.·10-s − 899.·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 1.53e3·15-s + 4.09e3·16-s − 3.88e4·17-s + 5.83e3·18-s + 2.40e3·19-s − 3.63e3·20-s + 9.26e3·21-s − 7.19e3·22-s + 1.06e5·23-s + 1.38e4·24-s − 7.49e4·25-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 − 0.202·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.143·10-s − 0.203·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.117·15-s + 0.250·16-s − 1.91·17-s + 0.235·18-s + 0.0805·19-s − 0.101·20-s + 0.218·21-s − 0.144·22-s + 1.82·23-s + 0.204·24-s − 0.958·25-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 | \( 1 + 56.7T + 7.81e4T^{2} \) |
| 11 | \( 1 + 899.T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.88e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.40e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.82e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.85e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.22e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.26e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.50e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.87e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.94e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.16e7T + 8.07e13T^{2} \) |
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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
−9.068571650987390221808852098882, −8.362015666965750397236328005962, −7.26259890694352644435136102816, −6.64781676366074957363477366502, −5.27091528430559779852272333158, −4.53618851491565665280085027304, −3.52745563909564851679608059800, −2.50560858660904269749806690551, −1.58655683938537044913434731101, 0,
1.58655683938537044913434731101, 2.50560858660904269749806690551, 3.52745563909564851679608059800, 4.53618851491565665280085027304, 5.27091528430559779852272333158, 6.64781676366074957363477366502, 7.26259890694352644435136102816, 8.362015666965750397236328005962, 9.068571650987390221808852098882