L(s) = 1 | − 2.77·2-s − 3-s + 5.67·4-s + 3.96·5-s + 2.77·6-s + 7-s − 10.1·8-s + 9-s − 10.9·10-s + 2.57·11-s − 5.67·12-s − 3.86·13-s − 2.77·14-s − 3.96·15-s + 16.8·16-s + 2.14·17-s − 2.77·18-s + 2.77·19-s + 22.4·20-s − 21-s − 7.12·22-s − 3.27·23-s + 10.1·24-s + 10.6·25-s + 10.7·26-s − 27-s + 5.67·28-s + ⋯ |
L(s) = 1 | − 1.95·2-s − 0.577·3-s + 2.83·4-s + 1.77·5-s + 1.13·6-s + 0.377·7-s − 3.59·8-s + 0.333·9-s − 3.47·10-s + 0.775·11-s − 1.63·12-s − 1.07·13-s − 0.740·14-s − 1.02·15-s + 4.21·16-s + 0.520·17-s − 0.652·18-s + 0.635·19-s + 5.02·20-s − 0.218·21-s − 1.51·22-s − 0.682·23-s + 2.07·24-s + 2.13·25-s + 2.10·26-s − 0.192·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 5 | \( 1 - 3.96T + 5T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 + 8.28T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 3.57T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 - 8.64T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.43378934615089846153055609854, −6.95491845881715151025703758095, −6.36398135595326247218423478734, −5.52067999583595277985708323149, −5.28795997397514142795886350855, −3.57816940520837113756768214742, −2.48194739403456746741945426758, −1.75225160700379003459331279582, −1.35518086958180082988547743247, 0,
1.35518086958180082988547743247, 1.75225160700379003459331279582, 2.48194739403456746741945426758, 3.57816940520837113756768214742, 5.28795997397514142795886350855, 5.52067999583595277985708323149, 6.36398135595326247218423478734, 6.95491845881715151025703758095, 7.43378934615089846153055609854