| L(s) = 1 | + 2.37·5-s − 4.37·7-s − 11-s − 0.372·13-s − 5.37·17-s − 0.627·19-s + 0.372·23-s + 0.627·25-s − 4.37·29-s − 6.37·31-s − 10.3·35-s + 8.74·37-s − 11.7·41-s + 1.74·43-s − 4.37·47-s + 12.1·49-s + 0.744·53-s − 2.37·55-s − 7·59-s + 2.37·61-s − 0.883·65-s + 3.74·67-s + 4·71-s − 12.1·73-s + 4.37·77-s + 6.37·79-s − 9.62·83-s + ⋯ |
| L(s) = 1 | + 1.06·5-s − 1.65·7-s − 0.301·11-s − 0.103·13-s − 1.30·17-s − 0.144·19-s + 0.0776·23-s + 0.125·25-s − 0.811·29-s − 1.14·31-s − 1.75·35-s + 1.43·37-s − 1.83·41-s + 0.266·43-s − 0.637·47-s + 1.73·49-s + 0.102·53-s − 0.319·55-s − 0.911·59-s + 0.303·61-s − 0.109·65-s + 0.457·67-s + 0.474·71-s − 1.41·73-s + 0.498·77-s + 0.716·79-s − 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 - 0.372T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 - 0.744T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 - 3.74T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 + 9.62T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−9.394158874294007273996782172605, −8.727770223738188031122951495364, −7.44786911605943622831691737688, −6.56769978400851266016617686027, −6.08502863571544441518884100940, −5.17202413054534468385232123754, −3.92564790152860563571748785695, −2.89236116533902573643406665998, −1.94950947461035593346148136329, 0,
1.94950947461035593346148136329, 2.89236116533902573643406665998, 3.92564790152860563571748785695, 5.17202413054534468385232123754, 6.08502863571544441518884100940, 6.56769978400851266016617686027, 7.44786911605943622831691737688, 8.727770223738188031122951495364, 9.394158874294007273996782172605
