| L(s) = 1 | − 2.73·3-s − 5-s − 0.732·7-s + 4.46·9-s − 2·11-s + 3.46·13-s + 2.73·15-s + 3.46·17-s − 0.535·19-s + 2·21-s + 6.19·23-s + 25-s − 3.99·27-s − 6.92·29-s − 5.46·31-s + 5.46·33-s + 0.732·35-s − 2·37-s − 9.46·39-s − 1.46·41-s − 5.26·43-s − 4.46·45-s + 3.26·47-s − 6.46·49-s − 9.46·51-s + 11.4·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.447·5-s − 0.276·7-s + 1.48·9-s − 0.603·11-s + 0.960·13-s + 0.705·15-s + 0.840·17-s − 0.122·19-s + 0.436·21-s + 1.29·23-s + 0.200·25-s − 0.769·27-s − 1.28·29-s − 0.981·31-s + 0.951·33-s + 0.123·35-s − 0.328·37-s − 1.51·39-s − 0.228·41-s − 0.803·43-s − 0.665·45-s + 0.476·47-s − 0.923·49-s − 1.32·51-s + 1.57·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−9.391738150394117520099174674125, −8.415944518628747317833252377998, −7.36927325093451665051001213038, −6.75435728538014509521987366642, −5.69423850032308280235077213487, −5.33118459531553052357886616137, −4.19837453503013577458556286505, −3.18634972588879378366548622359, −1.34206342860404764783195790078, 0,
1.34206342860404764783195790078, 3.18634972588879378366548622359, 4.19837453503013577458556286505, 5.33118459531553052357886616137, 5.69423850032308280235077213487, 6.75435728538014509521987366642, 7.36927325093451665051001213038, 8.415944518628747317833252377998, 9.391738150394117520099174674125
