| L(s) = 1 | + 0.495·2-s − 0.133·3-s − 1.75·4-s − 0.493·5-s − 0.0663·6-s + 1.67·7-s − 1.86·8-s − 2.98·9-s − 0.244·10-s + 11-s + 0.234·12-s + 5.28·13-s + 0.832·14-s + 0.0660·15-s + 2.58·16-s + 1.59·17-s − 1.47·18-s − 7.18·19-s + 0.865·20-s − 0.224·21-s + 0.495·22-s − 1.46·23-s + 0.248·24-s − 4.75·25-s + 2.61·26-s + 0.800·27-s − 2.94·28-s + ⋯ |
| L(s) = 1 | + 0.350·2-s − 0.0772·3-s − 0.877·4-s − 0.220·5-s − 0.0270·6-s + 0.634·7-s − 0.658·8-s − 0.994·9-s − 0.0773·10-s + 0.301·11-s + 0.0677·12-s + 1.46·13-s + 0.222·14-s + 0.0170·15-s + 0.646·16-s + 0.386·17-s − 0.348·18-s − 1.64·19-s + 0.193·20-s − 0.0489·21-s + 0.105·22-s − 0.304·23-s + 0.0508·24-s − 0.951·25-s + 0.513·26-s + 0.153·27-s − 0.556·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 0.495T + 2T^{2} \) |
| 3 | \( 1 + 0.133T + 3T^{2} \) |
| 5 | \( 1 + 0.493T + 5T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 0.0259T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−8.964719197856348460584202003437, −8.338514241746182197644350720163, −7.86195700601042733963391162064, −6.24831899479821314595349796393, −5.90721546520257766258554428492, −4.82594508144437505713164013015, −4.01407281005421859822456581474, −3.25426740188821848676216600792, −1.68019608002660471127652184625, 0,
1.68019608002660471127652184625, 3.25426740188821848676216600792, 4.01407281005421859822456581474, 4.82594508144437505713164013015, 5.90721546520257766258554428492, 6.24831899479821314595349796393, 7.86195700601042733963391162064, 8.338514241746182197644350720163, 8.964719197856348460584202003437
