| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s − 2·15-s + 6·17-s + 4·19-s + 23-s − 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s + 2·45-s − 8·47-s − 6·51-s + 6·53-s + 8·55-s − 4·57-s + 4·59-s + 10·61-s + 4·65-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 + 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.840·51-s + 0.824·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s + 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.442785838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.442785838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.33074294762652, −14.04200911692814, −13.39676803173139, −12.85291954802644, −12.44155975937681, −11.83475080396901, −11.25818025955695, −11.07678327043070, −10.17559761169382, −9.668699371536379, −9.532044774659776, −8.833505483951443, −8.164716531412897, −7.426896541943512, −7.074898371349522, −6.296759412054485, −5.836045795542731, −5.526126460724615, −4.894035168055092, −3.916534172314146, −3.677959024337596, −2.795931009725159, −1.902965433998598, −1.291122499144189, −0.7483982366653384,
0.7483982366653384, 1.291122499144189, 1.902965433998598, 2.795931009725159, 3.677959024337596, 3.916534172314146, 4.894035168055092, 5.526126460724615, 5.836045795542731, 6.296759412054485, 7.074898371349522, 7.426896541943512, 8.164716531412897, 8.833505483951443, 9.532044774659776, 9.668699371536379, 10.17559761169382, 11.07678327043070, 11.25818025955695, 11.83475080396901, 12.44155975937681, 12.85291954802644, 13.39676803173139, 14.04200911692814, 14.33074294762652
