| L(s) = 1 | + 0.517·2-s − 1.73·4-s + 3.15·5-s + 7-s − 1.93·8-s + 1.63·10-s + 1.34·11-s + 6.17·13-s + 0.517·14-s + 2.46·16-s − 0.162·17-s + 4.03·19-s − 5.47·20-s + 0.694·22-s + 9.14·23-s + 4.97·25-s + 3.19·26-s − 1.73·28-s + 3.08·29-s − 1.81·31-s + 5.13·32-s − 0.0841·34-s + 3.15·35-s − 1.51·37-s + 2.09·38-s − 6.10·40-s − 6.78·41-s + ⋯ |
| L(s) = 1 | + 0.365·2-s − 0.866·4-s + 1.41·5-s + 0.377·7-s − 0.682·8-s + 0.516·10-s + 0.404·11-s + 1.71·13-s + 0.138·14-s + 0.616·16-s − 0.0394·17-s + 0.926·19-s − 1.22·20-s + 0.148·22-s + 1.90·23-s + 0.995·25-s + 0.626·26-s − 0.327·28-s + 0.572·29-s − 0.326·31-s + 0.908·32-s − 0.0144·34-s + 0.533·35-s − 0.248·37-s + 0.339·38-s − 0.964·40-s − 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.502009366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.502009366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 + 0.162T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 - 9.14T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 6.78T + 41T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 - 0.216T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 9.56T + 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 7.71T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 7.78T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.021126356753149102322311515032, −6.87526277253252352409108202827, −6.34387797595214107312240573754, −5.53885517440251493999197957768, −5.23798705756006768999423566675, −4.38162789880629717957713928086, −3.49411695417586631775773512685, −2.86154787924404878400830482062, −1.55172794960242260602350354498, −1.01480445394391595422696921100,
1.01480445394391595422696921100, 1.55172794960242260602350354498, 2.86154787924404878400830482062, 3.49411695417586631775773512685, 4.38162789880629717957713928086, 5.23798705756006768999423566675, 5.53885517440251493999197957768, 6.34387797595214107312240573754, 6.87526277253252352409108202827, 8.021126356753149102322311515032
