| L(s) = 1 | + 2·7-s + 13-s + 2·19-s + 6·23-s − 8·31-s + 2·37-s − 6·41-s + 4·43-s − 3·49-s + 6·53-s − 14·61-s + 4·67-s + 4·73-s + 16·79-s − 12·83-s + 6·89-s + 2·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.277·13-s + 0.458·19-s + 1.25·23-s − 1.43·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 3/7·49-s + 0.824·53-s − 1.79·61-s + 0.488·67-s + 0.468·73-s + 1.80·79-s − 1.31·83-s + 0.635·89-s + 0.209·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.737848166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.737848166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.15181645825506, −12.68001559920833, −12.13356282128714, −11.71917543443622, −11.17331425936453, −10.78064489947326, −10.54267490924957, −9.653566836170754, −9.352100913466988, −8.875919983345959, −8.321924363936921, −7.880139032090804, −7.353332636881169, −6.936181497939468, −6.359327660970627, −5.728868309672894, −5.184820567183067, −4.919512293012944, −4.178331439901231, −3.673294148889767, −3.069167671228121, −2.486379379843560, −1.707015805331434, −1.290029096837981, −0.4838854506389896,
0.4838854506389896, 1.290029096837981, 1.707015805331434, 2.486379379843560, 3.069167671228121, 3.673294148889767, 4.178331439901231, 4.919512293012944, 5.184820567183067, 5.728868309672894, 6.359327660970627, 6.936181497939468, 7.353332636881169, 7.880139032090804, 8.321924363936921, 8.875919983345959, 9.352100913466988, 9.653566836170754, 10.54267490924957, 10.78064489947326, 11.17331425936453, 11.71917543443622, 12.13356282128714, 12.68001559920833, 13.15181645825506
