| L(s) = 1 | + 0.801·2-s − 1.35·4-s + 0.246·5-s + 2.35·7-s − 2.69·8-s + 0.198·10-s − 4.24·11-s + 1.89·14-s + 0.554·16-s − 2.15·17-s + 0.0881·19-s − 0.335·20-s − 3.40·22-s − 1.49·23-s − 4.93·25-s − 3.19·28-s − 4.63·29-s + 6.63·31-s + 5.82·32-s − 1.73·34-s + 0.582·35-s − 5.69·37-s + 0.0706·38-s − 0.664·40-s − 11.5·41-s − 0.295·43-s + 5.76·44-s + ⋯ |
| L(s) = 1 | + 0.567·2-s − 0.678·4-s + 0.110·5-s + 0.890·7-s − 0.951·8-s + 0.0626·10-s − 1.28·11-s + 0.505·14-s + 0.138·16-s − 0.523·17-s + 0.0202·19-s − 0.0749·20-s − 0.726·22-s − 0.311·23-s − 0.987·25-s − 0.604·28-s − 0.859·29-s + 1.19·31-s + 1.03·32-s − 0.296·34-s + 0.0983·35-s − 0.935·37-s + 0.0114·38-s − 0.105·40-s − 1.81·41-s − 0.0451·43-s + 0.868·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.801T + 2T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 - 0.0881T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 4.63T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 + 5.69T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.295T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 + 8.66T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 - 8.05T + 97T^{2} \) |
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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
−8.944735913564894074107163486264, −8.256002448862379376019189165165, −7.64643393869531211553791412970, −6.43954466659674096618197230360, −5.44807049899792278583441358927, −4.96480573827133245506737810255, −4.11282169508202190310335294525, −3.04579259481016993615149156303, −1.85717612077765170814572916069, 0,
1.85717612077765170814572916069, 3.04579259481016993615149156303, 4.11282169508202190310335294525, 4.96480573827133245506737810255, 5.44807049899792278583441358927, 6.43954466659674096618197230360, 7.64643393869531211553791412970, 8.256002448862379376019189165165, 8.944735913564894074107163486264
