| L(s) = 1 | − 2-s + 2.66·3-s + 4-s + 3.04·5-s − 2.66·6-s + 4.28·7-s − 8-s + 4.08·9-s − 3.04·10-s − 5.77·11-s + 2.66·12-s + 6.17·13-s − 4.28·14-s + 8.09·15-s + 16-s + 3.29·17-s − 4.08·18-s − 2.74·19-s + 3.04·20-s + 11.3·21-s + 5.77·22-s + 7.55·23-s − 2.66·24-s + 4.25·25-s − 6.17·26-s + 2.87·27-s + 4.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 0.5·4-s + 1.36·5-s − 1.08·6-s + 1.61·7-s − 0.353·8-s + 1.36·9-s − 0.962·10-s − 1.74·11-s + 0.768·12-s + 1.71·13-s − 1.14·14-s + 2.09·15-s + 0.250·16-s + 0.798·17-s − 0.962·18-s − 0.630·19-s + 0.680·20-s + 2.48·21-s + 1.23·22-s + 1.57·23-s − 0.543·24-s + 0.851·25-s − 1.21·26-s + 0.553·27-s + 0.808·28-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
| \(L(1)\) |
\(\approx\) |
\(3.912476210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.912476210\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 7.95T + 73T^{2} \) |
| 79 | \( 1 + 4.07T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 + 1.45T + 89T^{2} \) |
| 97 | \( 1 - 4.44T + 97T^{2} \) |
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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
−8.521575230913734045489121600061, −8.005926859409408215168886641520, −7.40255030963518796883436312055, −6.32670796129711676528512786492, −5.42453270096696965446574045155, −4.82507731303046178335719338355, −3.42706354232245250096193902166, −2.72673536455393911981756433719, −1.81193981453861027829629943543, −1.42363587186040688785107337438,
1.42363587186040688785107337438, 1.81193981453861027829629943543, 2.72673536455393911981756433719, 3.42706354232245250096193902166, 4.82507731303046178335719338355, 5.42453270096696965446574045155, 6.32670796129711676528512786492, 7.40255030963518796883436312055, 8.005926859409408215168886641520, 8.521575230913734045489121600061
