| L(s) = 1 | + 3-s + 9-s + 4·11-s + 6·13-s + 2·17-s − 4·19-s + 8·23-s + 27-s − 2·29-s + 4·33-s + 10·37-s + 6·39-s + 6·41-s − 4·43-s + 2·51-s − 6·53-s − 4·57-s + 4·59-s − 6·61-s + 4·67-s + 8·69-s − 8·71-s + 10·73-s + 81-s + 4·83-s − 2·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.960·39-s + 0.937·41-s − 0.609·43-s + 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.488·67-s + 0.963·69-s − 0.949·71-s + 1.17·73-s + 1/9·81-s + 0.439·83-s − 0.214·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.699662729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.699662729\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31508379381776, −13.89526827647895, −13.29894000538157, −12.86758428338016, −12.53524797492079, −11.63141450063544, −11.28594800364901, −10.85506754917834, −10.27503806206922, −9.481668178293803, −9.138201192684047, −8.771463859551408, −8.122284022329673, −7.705188270599338, −6.875471343397471, −6.476897959143961, −6.011342564916520, −5.301987299563936, −4.427072856407022, −4.069563392223029, −3.403908307413966, −2.938893464297023, −2.020265449799034, −1.300367912137640, −0.8078845242002148,
0.8078845242002148, 1.300367912137640, 2.020265449799034, 2.938893464297023, 3.403908307413966, 4.069563392223029, 4.427072856407022, 5.301987299563936, 6.011342564916520, 6.476897959143961, 6.875471343397471, 7.705188270599338, 8.122284022329673, 8.771463859551408, 9.138201192684047, 9.481668178293803, 10.27503806206922, 10.85506754917834, 11.28594800364901, 11.63141450063544, 12.53524797492079, 12.86758428338016, 13.29894000538157, 13.89526827647895, 14.31508379381776
