L(s) = 1 | − 1.17·2-s − 0.620·4-s − 3.75·5-s − 7-s + 3.07·8-s + 4.40·10-s + 0.236·11-s + 2.34·13-s + 1.17·14-s − 2.37·16-s + 5.49·17-s + 1.29·19-s + 2.32·20-s − 0.278·22-s − 4.33·23-s + 9.08·25-s − 2.75·26-s + 0.620·28-s − 3.35·29-s + 10.2·31-s − 3.36·32-s − 6.45·34-s + 3.75·35-s + 5.67·37-s − 1.52·38-s − 11.5·40-s − 5.91·41-s + ⋯ |
L(s) = 1 | − 0.830·2-s − 0.310·4-s − 1.67·5-s − 0.377·7-s + 1.08·8-s + 1.39·10-s + 0.0714·11-s + 0.649·13-s + 0.313·14-s − 0.593·16-s + 1.33·17-s + 0.297·19-s + 0.520·20-s − 0.0593·22-s − 0.904·23-s + 1.81·25-s − 0.539·26-s + 0.117·28-s − 0.622·29-s + 1.83·31-s − 0.594·32-s − 1.10·34-s + 0.634·35-s + 0.932·37-s − 0.247·38-s − 1.82·40-s − 0.924·41-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
\(L(1)\) |
\(\approx\) |
\(0.6683967538\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6683967538\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 23 | \( 1 + 4.33T + 23T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 - 7.58T + 43T^{2} \) |
| 47 | \( 1 + 6.69T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 2.42T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 + 1.63T + 97T^{2} \) |
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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
−7.933700011057163648854085462614, −7.49606031986771354031348210610, −6.71797574610908431716821036473, −5.79339512904984803875536253916, −4.90384180214890505965886833853, −4.04549097587795955871606886290, −3.73118139580341346045211126486, −2.77943720671810537622257272560, −1.27424327920338525565832402569, −0.53489181941884212274930716605,
0.53489181941884212274930716605, 1.27424327920338525565832402569, 2.77943720671810537622257272560, 3.73118139580341346045211126486, 4.04549097587795955871606886290, 4.90384180214890505965886833853, 5.79339512904984803875536253916, 6.71797574610908431716821036473, 7.49606031986771354031348210610, 7.933700011057163648854085462614