| L(s) = 1 | + 3-s − 5-s + 9-s − 2·13-s − 15-s + 2·17-s + 4·19-s − 8·23-s + 25-s + 27-s − 2·29-s + 4·31-s − 6·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s + 2·51-s + 6·53-s + 4·57-s − 6·61-s + 2·65-s − 4·67-s − 8·69-s − 8·71-s − 10·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.768·61-s + 0.248·65-s − 0.488·67-s − 0.963·69-s − 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.77480231197802, −15.14965673305237, −14.58914476469922, −14.12301327638275, −13.71728370619541, −13.05000401829288, −12.40060568606281, −11.89374774857177, −11.62915701973964, −10.63268843646086, −10.22203228337653, −9.663255521930231, −9.083463592252782, −8.484234629002552, −7.777563183196817, −7.528708506913781, −6.885696383324599, −5.987962798079155, −5.522981146561025, −4.617054714357309, −4.120896529996861, −3.396465676103011, −2.781906735340971, −2.001713951546988, −1.120394346087134, 0,
1.120394346087134, 2.001713951546988, 2.781906735340971, 3.396465676103011, 4.120896529996861, 4.617054714357309, 5.522981146561025, 5.987962798079155, 6.885696383324599, 7.528708506913781, 7.777563183196817, 8.484234629002552, 9.083463592252782, 9.663255521930231, 10.22203228337653, 10.63268843646086, 11.62915701973964, 11.89374774857177, 12.40060568606281, 13.05000401829288, 13.71728370619541, 14.12301327638275, 14.58914476469922, 15.14965673305237, 15.77480231197802
