| L(s) = 1 | + 1.15·3-s + 1.07·5-s + 0.238·7-s − 1.67·9-s + 2.84·11-s − 1.31·13-s + 1.23·15-s − 4.47·17-s − 19-s + 0.274·21-s + 1.94·23-s − 3.84·25-s − 5.38·27-s − 8.83·29-s − 6.64·31-s + 3.27·33-s + 0.256·35-s + 4.06·37-s − 1.51·39-s − 2.78·41-s + 1.21·43-s − 1.80·45-s − 10.2·47-s − 6.94·49-s − 5.15·51-s + 8.36·53-s + 3.05·55-s + ⋯ |
| L(s) = 1 | + 0.664·3-s + 0.481·5-s + 0.0901·7-s − 0.557·9-s + 0.857·11-s − 0.364·13-s + 0.319·15-s − 1.08·17-s − 0.229·19-s + 0.0599·21-s + 0.405·23-s − 0.768·25-s − 1.03·27-s − 1.64·29-s − 1.19·31-s + 0.569·33-s + 0.0433·35-s + 0.668·37-s − 0.242·39-s − 0.434·41-s + 0.185·43-s − 0.268·45-s − 1.49·47-s − 0.991·49-s − 0.721·51-s + 1.14·53-s + 0.412·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 - 0.238T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 23 | \( 1 - 1.94T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 8.36T + 53T^{2} \) |
| 59 | \( 1 + 9.25T + 59T^{2} \) |
| 61 | \( 1 - 4.66T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 - 2.78T + 71T^{2} \) |
| 73 | \( 1 + 0.134T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 + 0.247T + 97T^{2} \) |
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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
−7.962857908910403399957204590128, −7.25606560106997614023235993967, −6.44610313283125018610110752250, −5.78954539014891677453167620444, −4.96933771150360799676890170128, −4.00299042290785335255413826516, −3.33566021420691983778385486717, −2.27831310397930921333534679448, −1.70979024365260129192829441671, 0,
1.70979024365260129192829441671, 2.27831310397930921333534679448, 3.33566021420691983778385486717, 4.00299042290785335255413826516, 4.96933771150360799676890170128, 5.78954539014891677453167620444, 6.44610313283125018610110752250, 7.25606560106997614023235993967, 7.962857908910403399957204590128
