| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s + 4·11-s − 2·13-s + 15-s + 2·17-s + 2·19-s + 4·21-s + 6·23-s + 25-s + 27-s + 6·29-s + 10·31-s + 4·33-s + 4·35-s + 37-s − 2·39-s − 2·41-s + 2·43-s + 45-s − 4·47-s + 9·49-s + 2·51-s − 10·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.696·33-s + 0.676·35-s + 0.164·37-s − 0.320·39-s − 0.312·41-s + 0.304·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.769612535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.769612535\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.78420649580200, −14.33658224002751, −14.06128285023708, −13.59192714002902, −12.85000221769429, −12.22264709013325, −11.76041268826155, −11.33596669202135, −10.69208619350121, −10.02577496373749, −9.596407414633705, −8.989563350802567, −8.444134597975370, −8.010173212980914, −7.420871915068969, −6.726114463329648, −6.302032273487651, −5.327781435813574, −4.839561001543782, −4.445787764028652, −3.579873503873467, −2.845144907049260, −2.224706311360347, −1.286069114025453, −1.057428183513603,
1.057428183513603, 1.286069114025453, 2.224706311360347, 2.845144907049260, 3.579873503873467, 4.445787764028652, 4.839561001543782, 5.327781435813574, 6.302032273487651, 6.726114463329648, 7.420871915068969, 8.010173212980914, 8.444134597975370, 8.989563350802567, 9.596407414633705, 10.02577496373749, 10.69208619350121, 11.33596669202135, 11.76041268826155, 12.22264709013325, 12.85000221769429, 13.59192714002902, 14.06128285023708, 14.33658224002751, 14.78420649580200
