| L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 6·17-s + 19-s + 4·23-s − 25-s + 27-s − 2·29-s − 4·31-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s + 2·45-s + 4·47-s − 7·49-s − 6·51-s − 10·53-s + 8·55-s + 57-s − 12·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 0.583·47-s − 49-s − 0.840·51-s − 1.37·53-s + 1.07·55-s + 0.132·57-s − 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.388472434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.388472434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.786869452003372550002449800084, −9.259741991745774642651687902060, −8.686112233403729699672924385749, −7.54920325839787930805373575681, −6.57061735757055000239038218298, −5.95928356394221873345303919204, −4.65672229415455481414222119791, −3.72816659383691276241941645312, −2.47746893642927435624182028712, −1.41228664440352179477085709942,
1.41228664440352179477085709942, 2.47746893642927435624182028712, 3.72816659383691276241941645312, 4.65672229415455481414222119791, 5.95928356394221873345303919204, 6.57061735757055000239038218298, 7.54920325839787930805373575681, 8.686112233403729699672924385749, 9.259741991745774642651687902060, 9.786869452003372550002449800084
