| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 4·11-s + 2·12-s + 15-s + 4·16-s − 5·17-s − 5·19-s + 2·20-s − 21-s + 23-s + 25-s − 27-s − 2·28-s + 29-s + 8·31-s − 4·33-s − 35-s − 2·36-s − 2·41-s − 4·43-s − 8·44-s − 45-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 0.258·15-s + 16-s − 1.21·17-s − 1.14·19-s + 0.447·20-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.185·29-s + 1.43·31-s − 0.696·33-s − 0.169·35-s − 1/3·36-s − 0.312·41-s − 0.609·43-s − 1.20·44-s − 0.149·45-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−17.09914420746369, −16.63046003621578, −15.57066539738424, −15.32068776143596, −14.62840040609125, −13.94896765809857, −13.61325671478171, −12.65034867082479, −12.47569725972473, −11.63194342271892, −11.17632376683816, −10.56118747993511, −9.823358967540153, −9.189149903269001, −8.599318667441114, −8.163030892600700, −7.284556363453506, −6.401097965241645, −6.191521593411092, −4.935865977001431, −4.604732894046005, −4.062992798324027, −3.243867105306771, −1.969940377002474, −1.002001059356334, 0,
1.002001059356334, 1.969940377002474, 3.243867105306771, 4.062992798324027, 4.604732894046005, 4.935865977001431, 6.191521593411092, 6.401097965241645, 7.284556363453506, 8.163030892600700, 8.599318667441114, 9.189149903269001, 9.823358967540153, 10.56118747993511, 11.17632376683816, 11.63194342271892, 12.47569725972473, 12.65034867082479, 13.61325671478171, 13.94896765809857, 14.62840040609125, 15.32068776143596, 15.57066539738424, 16.63046003621578, 17.09914420746369
