| L(s) = 1 | + 0.716·2-s − 0.503·3-s − 1.48·4-s − 0.847·5-s − 0.360·6-s + 1.18·7-s − 2.49·8-s − 2.74·9-s − 0.606·10-s − 1.22·11-s + 0.748·12-s + 4.75·13-s + 0.848·14-s + 0.426·15-s + 1.18·16-s − 1.01·17-s − 1.96·18-s + 6.87·19-s + 1.26·20-s − 0.596·21-s − 0.875·22-s + 4.16·23-s + 1.25·24-s − 4.28·25-s + 3.40·26-s + 2.89·27-s − 1.76·28-s + ⋯ |
| L(s) = 1 | + 0.506·2-s − 0.290·3-s − 0.743·4-s − 0.378·5-s − 0.147·6-s + 0.447·7-s − 0.882·8-s − 0.915·9-s − 0.191·10-s − 0.368·11-s + 0.216·12-s + 1.31·13-s + 0.226·14-s + 0.110·15-s + 0.296·16-s − 0.245·17-s − 0.463·18-s + 1.57·19-s + 0.281·20-s − 0.130·21-s − 0.186·22-s + 0.867·23-s + 0.256·24-s − 0.856·25-s + 0.667·26-s + 0.556·27-s − 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
| good | 2 | \( 1 - 0.716T + 2T^{2} \) |
| 3 | \( 1 + 0.503T + 3T^{2} \) |
| 5 | \( 1 + 0.847T + 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 - 6.87T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 5.30T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 4.44T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.359710737267378294364148952122, −8.000662505054688476626649373649, −6.76107753947158156819683990527, −5.96524957634281382896851688255, −5.17405325950292888831397651427, −4.75528997270574559023632351280, −3.47695983027094321507301728584, −3.14323156040330315133882492435, −1.37861662177753163780039308687, 0,
1.37861662177753163780039308687, 3.14323156040330315133882492435, 3.47695983027094321507301728584, 4.75528997270574559023632351280, 5.17405325950292888831397651427, 5.96524957634281382896851688255, 6.76107753947158156819683990527, 8.000662505054688476626649373649, 8.359710737267378294364148952122
