L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 4·11-s + 2·13-s + 15-s − 2·17-s − 21-s + 4·23-s + 25-s − 27-s + 2·29-s − 8·31-s − 4·33-s − 35-s + 10·37-s − 2·39-s + 2·41-s − 4·43-s − 45-s − 4·47-s + 49-s + 2·51-s + 10·53-s − 4·55-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.590159989\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590159989\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.779163763171768671397090967568, −7.77808251491123488770635639195, −7.07796429504214504141120182319, −6.38464894774348283654875163447, −5.66903868936721114141625149330, −4.67194349957721396496507728240, −4.09539147032376430364048977349, −3.18788712742949030443501949771, −1.81230309300758118633280231628, −0.810232776946839969953606488595,
0.810232776946839969953606488595, 1.81230309300758118633280231628, 3.18788712742949030443501949771, 4.09539147032376430364048977349, 4.67194349957721396496507728240, 5.66903868936721114141625149330, 6.38464894774348283654875163447, 7.07796429504214504141120182319, 7.77808251491123488770635639195, 8.779163763171768671397090967568