| L(s) = 1 | − 5.19i·3-s + 6·5-s − 62.3i·7-s − 27·9-s − 187. i·11-s − 86·13-s − 31.1i·15-s + 426·17-s − 62.3i·19-s − 324·21-s + 748. i·23-s − 589·25-s + 140. i·27-s + 1.18e3·29-s + 1.55e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.239·5-s − 1.27i·7-s − 0.333·9-s − 1.54i·11-s − 0.508·13-s − 0.138i·15-s + 1.47·17-s − 0.172i·19-s − 0.734·21-s + 1.41i·23-s − 0.942·25-s + 0.192i·27-s + 1.40·29-s + 1.62i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.01974 - 1.01974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01974 - 1.01974i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 - 6T + 625T^{2} \) |
| 7 | \( 1 + 62.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 187. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 86T + 2.85e4T^{2} \) |
| 17 | \( 1 - 426T + 8.35e4T^{2} \) |
| 19 | \( 1 + 62.3iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 748. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.18e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.55e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 430T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.25e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.68e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 374. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.60e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.55e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.11e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.30e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.99e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.06e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.54e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.16e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.04e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 2.94e3T + 8.85e7T^{2} \) |
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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
−13.99674766529058251096525996969, −13.86924617555421987527962627285, −12.35400131475003144023861275541, −11.08410735332800373285155252329, −9.911278195966264303669118365165, −8.226501656210905022532656375581, −7.09138556043584818311730088930, −5.57794918279561849813948839302, −3.42841110485394955965120713675, −0.962472847177862525655787463656,
2.46075016701042556698623068330, 4.63887968615779976205185011438, 6.00217194436756973025855971757, 7.87186580605940306520826360810, 9.390896301329452375271742999677, 10.14836976428060325075866909605, 11.89890521089086699520929490451, 12.59975500553505287172079269396, 14.44044932249046000372628729897, 15.08648996790147396481997118942
