| L(s) = 1 | − 0.468·2-s − 1.50·3-s − 1.78·4-s + 3.60·5-s + 0.707·6-s + 0.116·7-s + 1.77·8-s − 0.724·9-s − 1.69·10-s − 3.89·11-s + 2.68·12-s − 2.52·13-s − 0.0545·14-s − 5.44·15-s + 2.72·16-s − 3.22·17-s + 0.339·18-s + 5.57·19-s − 6.42·20-s − 0.175·21-s + 1.82·22-s + 3.18·23-s − 2.67·24-s + 8.02·25-s + 1.18·26-s + 5.61·27-s − 0.207·28-s + ⋯ |
| L(s) = 1 | − 0.331·2-s − 0.870·3-s − 0.890·4-s + 1.61·5-s + 0.288·6-s + 0.0439·7-s + 0.626·8-s − 0.241·9-s − 0.535·10-s − 1.17·11-s + 0.775·12-s − 0.701·13-s − 0.0145·14-s − 1.40·15-s + 0.682·16-s − 0.782·17-s + 0.0800·18-s + 1.27·19-s − 1.43·20-s − 0.0382·21-s + 0.389·22-s + 0.663·23-s − 0.545·24-s + 1.60·25-s + 0.232·26-s + 1.08·27-s − 0.0391·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8072798865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8072798865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 0.468T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 7 | \( 1 - 0.116T + 7T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 + 9.54T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 - 5.69T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 1.30T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.891543290807255655790429612873, −7.04123952682401973469142180991, −6.36732931211391156293793899874, −5.37968648847912908975142610879, −5.28222831898314913983465245109, −4.80721258391680765502578866946, −3.41693042700367011398816296517, −2.51705282550400497898163588985, −1.62893431981376837273386359716, −0.49459004568494754477721820261,
0.49459004568494754477721820261, 1.62893431981376837273386359716, 2.51705282550400497898163588985, 3.41693042700367011398816296517, 4.80721258391680765502578866946, 5.28222831898314913983465245109, 5.37968648847912908975142610879, 6.36732931211391156293793899874, 7.04123952682401973469142180991, 7.891543290807255655790429612873
