L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s − 7-s + 8-s − 2·9-s − 3·10-s + 3·11-s − 12-s − 2·13-s − 14-s + 3·15-s + 16-s + 8·17-s − 2·18-s − 5·19-s − 3·20-s + 21-s + 3·22-s + 5·23-s − 24-s + 4·25-s − 2·26-s + 5·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.904·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.94·17-s − 0.471·18-s − 1.14·19-s − 0.670·20-s + 0.218·21-s + 0.639·22-s + 1.04·23-s − 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.47472394362643551620700778190, −7.00022062598338388243054161295, −6.25224370283616048267129182885, −5.47048063871350917711019208159, −4.89274410361393711574174807666, −3.91323227455150479899506813933, −3.49724381023122979762231614365, −2.64533094001853263186898805892, −1.17271125004645804010838955785, 0,
1.17271125004645804010838955785, 2.64533094001853263186898805892, 3.49724381023122979762231614365, 3.91323227455150479899506813933, 4.89274410361393711574174807666, 5.47048063871350917711019208159, 6.25224370283616048267129182885, 7.00022062598338388243054161295, 7.47472394362643551620700778190