L(s) = 1 | − 2-s + 1.04·3-s + 4-s − 1.51·5-s − 1.04·6-s − 0.313·7-s − 8-s − 1.91·9-s + 1.51·10-s − 3.91·11-s + 1.04·12-s + 1.10·13-s + 0.313·14-s − 1.58·15-s + 16-s + 4.20·17-s + 1.91·18-s + 5.34·19-s − 1.51·20-s − 0.327·21-s + 3.91·22-s + 7.60·23-s − 1.04·24-s − 2.70·25-s − 1.10·26-s − 5.12·27-s − 0.313·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.602·3-s + 0.5·4-s − 0.677·5-s − 0.426·6-s − 0.118·7-s − 0.353·8-s − 0.636·9-s + 0.479·10-s − 1.18·11-s + 0.301·12-s + 0.307·13-s + 0.0838·14-s − 0.408·15-s + 0.250·16-s + 1.01·17-s + 0.450·18-s + 1.22·19-s − 0.338·20-s − 0.0714·21-s + 0.835·22-s + 1.58·23-s − 0.213·24-s − 0.541·25-s − 0.217·26-s − 0.986·27-s − 0.0592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 0.313T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 + 0.0679T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 - 2.88T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 1.87T + 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 0.929T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 1.29T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−8.165665017564011924542005692262, −7.43264666999062378537571876595, −7.12160729963521204760483256332, −5.59985724243854422675296911371, −5.44694590594897807966682134897, −3.99217001299782884320182769170, −3.12319158575475687216286896915, −2.67506796639323647912816054253, −1.28278967339892228712069361842, 0,
1.28278967339892228712069361842, 2.67506796639323647912816054253, 3.12319158575475687216286896915, 3.99217001299782884320182769170, 5.44694590594897807966682134897, 5.59985724243854422675296911371, 7.12160729963521204760483256332, 7.43264666999062378537571876595, 8.165665017564011924542005692262