| L(s) = 1 | + 2.88·3-s − 3.33·5-s + 0.0162·7-s − 18.6·9-s − 38.3·11-s − 47.9·13-s − 9.62·15-s + 23.3·17-s + 80.2·19-s + 0.0469·21-s − 161.·23-s − 113.·25-s − 131.·27-s − 29·29-s + 43.4·31-s − 110.·33-s − 0.0542·35-s + 154.·37-s − 138.·39-s + 330.·41-s − 271.·43-s + 62.2·45-s − 314.·47-s − 342.·49-s + 67.3·51-s − 105.·53-s + 127.·55-s + ⋯ |
| L(s) = 1 | + 0.555·3-s − 0.298·5-s + 0.000879·7-s − 0.691·9-s − 1.05·11-s − 1.02·13-s − 0.165·15-s + 0.332·17-s + 0.968·19-s + 0.000487·21-s − 1.46·23-s − 0.911·25-s − 0.939·27-s − 0.185·29-s + 0.251·31-s − 0.582·33-s − 0.000262·35-s + 0.687·37-s − 0.567·39-s + 1.26·41-s − 0.963·43-s + 0.206·45-s − 0.974·47-s − 0.999·49-s + 0.184·51-s − 0.273·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 2.88T + 27T^{2} \) |
| 5 | \( 1 + 3.33T + 125T^{2} \) |
| 7 | \( 1 - 0.0162T + 343T^{2} \) |
| 11 | \( 1 + 38.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 43.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 419.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 240.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 252.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 842.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 125.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 730.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.74e3T + 9.12e5T^{2} \) |
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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
−11.39660296612057317738989594020, −10.12302747574042689235489727809, −9.391414345853392886675456689418, −7.970297550895990221160060196360, −7.71170200313619935103664782311, −6.02098460231851183120496830961, −4.92084309024214025258196428414, −3.41484096226827148790926224139, −2.29131263198630856565433557600, 0,
2.29131263198630856565433557600, 3.41484096226827148790926224139, 4.92084309024214025258196428414, 6.02098460231851183120496830961, 7.71170200313619935103664782311, 7.970297550895990221160060196360, 9.391414345853392886675456689418, 10.12302747574042689235489727809, 11.39660296612057317738989594020
