| L(s) = 1 | − 2-s − 2.37·3-s + 4-s − 3.32·5-s + 2.37·6-s + 3.18·7-s − 8-s + 2.64·9-s + 3.32·10-s − 3.12·11-s − 2.37·12-s − 4.61·13-s − 3.18·14-s + 7.91·15-s + 16-s + 5.18·17-s − 2.64·18-s + 3.13·19-s − 3.32·20-s − 7.56·21-s + 3.12·22-s + 5.95·23-s + 2.37·24-s + 6.08·25-s + 4.61·26-s + 0.841·27-s + 3.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.37·3-s + 0.5·4-s − 1.48·5-s + 0.970·6-s + 1.20·7-s − 0.353·8-s + 0.881·9-s + 1.05·10-s − 0.942·11-s − 0.685·12-s − 1.27·13-s − 0.850·14-s + 2.04·15-s + 0.250·16-s + 1.25·17-s − 0.623·18-s + 0.719·19-s − 0.744·20-s − 1.65·21-s + 0.666·22-s + 1.24·23-s + 0.485·24-s + 1.21·25-s + 0.904·26-s + 0.161·27-s + 0.601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4766379318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4766379318\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 - 1.91T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 5.59T + 59T^{2} \) |
| 61 | \( 1 + 1.78T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 2.57T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 5.61T + 97T^{2} \) |
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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
−7.81222339910386627985820864536, −7.31702014421326993204107171967, −6.72436835023430051329026241068, −5.49321497338335878337806120406, −5.11519099473781313151215890961, −4.64433916041102609604814597583, −3.49986460585785814919453727224, −2.61318782383449582373632497215, −1.26516067451688030847580514835, −0.46416902132102933982996272678,
0.46416902132102933982996272678, 1.26516067451688030847580514835, 2.61318782383449582373632497215, 3.49986460585785814919453727224, 4.64433916041102609604814597583, 5.11519099473781313151215890961, 5.49321497338335878337806120406, 6.72436835023430051329026241068, 7.31702014421326993204107171967, 7.81222339910386627985820864536
