| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 6·17-s + 4·19-s + 21-s + 23-s − 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s − 2·35-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 6·51-s − 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-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 + 0.218·21-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 − 0.338·35-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 + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 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
−15.51488372963165, −15.04254790988320, −13.90307886349619, −13.80525256539370, −13.29212580274043, −12.83058794746369, −12.23123110991322, −11.53114982054522, −11.13080314108051, −10.43901340667491, −10.03885322377946, −9.672219887256609, −8.785198622154931, −8.454783005975069, −7.644635717861100, −6.920208246854855, −6.507915866615559, −5.911330512676735, −5.296494437975712, −4.901057945466802, −4.097379959984436, −3.243305531865748, −2.538101165401875, −1.914522607705434, −0.9653886780361722, 0,
0.9653886780361722, 1.914522607705434, 2.538101165401875, 3.243305531865748, 4.097379959984436, 4.901057945466802, 5.296494437975712, 5.911330512676735, 6.507915866615559, 6.920208246854855, 7.644635717861100, 8.454783005975069, 8.785198622154931, 9.672219887256609, 10.03885322377946, 10.43901340667491, 11.13080314108051, 11.53114982054522, 12.23123110991322, 12.83058794746369, 13.29212580274043, 13.80525256539370, 13.90307886349619, 15.04254790988320, 15.51488372963165
