| L(s) = 1 | − 5-s + 4·7-s + 4·11-s − 6·13-s + 2·17-s + 4·19-s + 23-s + 25-s − 10·29-s − 8·31-s − 4·35-s − 2·37-s − 2·41-s + 8·43-s + 9·49-s − 6·53-s − 4·55-s − 6·61-s + 6·65-s − 8·67-s + 4·71-s + 10·73-s + 16·77-s + 16·79-s − 12·83-s − 2·85-s + 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.85·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s − 0.312·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 0.768·61-s + 0.744·65-s − 0.977·67-s + 0.474·71-s + 1.17·73-s + 1.82·77-s + 1.80·79-s − 1.31·83-s − 0.216·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.58078656803548, −14.06350407107325, −13.67931108188482, −12.69570708177406, −12.39167184990701, −11.90113354013669, −11.44958877619347, −11.05041417136945, −10.57911668132563, −9.686884792522089, −9.340892641561484, −8.964658215097944, −8.139300978524047, −7.570001167383601, −7.446203401260732, −6.867842168241906, −5.959161435078622, −5.302368973820573, −5.017938815997010, −4.338403478214680, −3.777994029979201, −3.182658008406126, −2.213019579651758, −1.717320390108410, −1.034229409321773, 0,
1.034229409321773, 1.717320390108410, 2.213019579651758, 3.182658008406126, 3.777994029979201, 4.338403478214680, 5.017938815997010, 5.302368973820573, 5.959161435078622, 6.867842168241906, 7.446203401260732, 7.570001167383601, 8.139300978524047, 8.964658215097944, 9.340892641561484, 9.686884792522089, 10.57911668132563, 11.05041417136945, 11.44958877619347, 11.90113354013669, 12.39167184990701, 12.69570708177406, 13.67931108188482, 14.06350407107325, 14.58078656803548
