L(s) = 1 | − 1.32·2-s − 0.255·4-s − 0.125·5-s + 1.14·7-s + 2.97·8-s + 0.165·10-s + 11-s − 2.50·13-s − 1.51·14-s − 3.42·16-s − 3.85·17-s + 6.27·19-s + 0.0319·20-s − 1.32·22-s + 3.82·23-s − 4.98·25-s + 3.30·26-s − 0.292·28-s + 1.92·29-s − 3.36·31-s − 1.43·32-s + 5.09·34-s − 0.143·35-s − 4.87·37-s − 8.29·38-s − 0.372·40-s + 12.3·41-s + ⋯ |
L(s) = 1 | − 0.933·2-s − 0.127·4-s − 0.0559·5-s + 0.433·7-s + 1.05·8-s + 0.0522·10-s + 0.301·11-s − 0.694·13-s − 0.404·14-s − 0.855·16-s − 0.934·17-s + 1.44·19-s + 0.00714·20-s − 0.281·22-s + 0.797·23-s − 0.996·25-s + 0.648·26-s − 0.0553·28-s + 0.356·29-s − 0.603·31-s − 0.253·32-s + 0.872·34-s − 0.0242·35-s − 0.800·37-s − 1.34·38-s − 0.0589·40-s + 1.92·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 + 0.125T + 5T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 6.57T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 + 8.55T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 9.44T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.61242960002150766568093683903, −7.42763704775216258763768174990, −6.52235441595635742878156722305, −5.47156766976465547623581344162, −4.82431053511558601316681579286, −4.16120434664477626443187836175, −3.13438324830162280986225731917, −2.02537223009966857607732510646, −1.17534428993669592335549457052, 0,
1.17534428993669592335549457052, 2.02537223009966857607732510646, 3.13438324830162280986225731917, 4.16120434664477626443187836175, 4.82431053511558601316681579286, 5.47156766976465547623581344162, 6.52235441595635742878156722305, 7.42763704775216258763768174990, 7.61242960002150766568093683903