| L(s) = 1 | − 3·7-s − 5·11-s − 13-s + 5·17-s + 4·19-s + 2·23-s − 9·29-s − 3·31-s + 10·37-s + 12·41-s + 2·43-s + 9·47-s + 2·49-s + 9·53-s − 3·59-s + 7·61-s + 9·67-s − 10·73-s + 15·77-s + 10·79-s − 83-s + 4·89-s + 3·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.50·11-s − 0.277·13-s + 1.21·17-s + 0.917·19-s + 0.417·23-s − 1.67·29-s − 0.538·31-s + 1.64·37-s + 1.87·41-s + 0.304·43-s + 1.31·47-s + 2/7·49-s + 1.23·53-s − 0.390·59-s + 0.896·61-s + 1.09·67-s − 1.17·73-s + 1.70·77-s + 1.12·79-s − 0.109·83-s + 0.423·89-s + 0.314·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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
| \(L(1)\) |
\(\approx\) |
\(2.099363989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099363989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.14578361807559, −12.75117521739907, −12.34207826725171, −11.73185836363097, −11.20063482764018, −10.71862415971325, −10.29115423511823, −9.707951544101216, −9.450459004124340, −9.062341155111755, −8.209795736675674, −7.741101234160691, −7.351609910010415, −7.107292045495906, −6.135604311140523, −5.739663453924521, −5.471574518731501, −4.871872959722383, −4.056680990230287, −3.628829934728895, −2.953241148386552, −2.635421309866830, −2.003069049272999, −0.9024160729569821, −0.5163414757519473,
0.5163414757519473, 0.9024160729569821, 2.003069049272999, 2.635421309866830, 2.953241148386552, 3.628829934728895, 4.056680990230287, 4.871872959722383, 5.471574518731501, 5.739663453924521, 6.135604311140523, 7.107292045495906, 7.351609910010415, 7.741101234160691, 8.209795736675674, 9.062341155111755, 9.450459004124340, 9.707951544101216, 10.29115423511823, 10.71862415971325, 11.20063482764018, 11.73185836363097, 12.34207826725171, 12.75117521739907, 13.14578361807559
