| L(s) = 1 | − 3-s − 4·7-s + 9-s − 4·11-s + 13-s − 6·17-s + 4·21-s + 4·23-s − 27-s + 6·29-s − 8·31-s + 4·33-s − 2·37-s − 39-s + 10·41-s − 4·43-s − 8·47-s + 9·49-s + 6·51-s − 2·53-s − 4·59-s − 14·61-s − 4·63-s − 12·67-s − 4·69-s − 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s − 0.520·59-s − 1.79·61-s − 0.503·63-s − 1.46·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.89525400683498, −14.22320179328799, −13.40577355290417, −13.22216622146172, −12.89470211392599, −12.33376247970666, −11.82183181415631, −11.00337290068401, −10.73820793415425, −10.38144719294556, −9.619264762196638, −9.201626119112376, −8.786639563315111, −7.958546737946582, −7.433724926725977, −6.789176126486129, −6.427930021854941, −5.929097382637909, −5.271335751926727, −4.700437029312747, −4.125240763369794, −3.256927308669433, −2.887227358816682, −2.161682445267374, −1.209592659864813, 0, 0,
1.209592659864813, 2.161682445267374, 2.887227358816682, 3.256927308669433, 4.125240763369794, 4.700437029312747, 5.271335751926727, 5.929097382637909, 6.427930021854941, 6.789176126486129, 7.433724926725977, 7.958546737946582, 8.786639563315111, 9.201626119112376, 9.619264762196638, 10.38144719294556, 10.73820793415425, 11.00337290068401, 11.82183181415631, 12.33376247970666, 12.89470211392599, 13.22216622146172, 13.40577355290417, 14.22320179328799, 14.89525400683498
