| L(s) = 1 | − 3.80i·3-s − 10.4·5-s + 18.5i·7-s + 66.5·9-s − 224. i·11-s − 312.·13-s + 39.8i·15-s − 321.·17-s + 199. i·19-s + 70.4·21-s − 450. i·23-s − 515.·25-s − 561. i·27-s − 807.·29-s − 141. i·31-s + ⋯ |
| L(s) = 1 | − 0.422i·3-s − 0.418·5-s + 0.377i·7-s + 0.821·9-s − 1.85i·11-s − 1.85·13-s + 0.177i·15-s − 1.11·17-s + 0.552i·19-s + 0.159·21-s − 0.851i·23-s − 0.824·25-s − 0.769i·27-s − 0.959·29-s − 0.147i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.191633 - 0.715184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191633 - 0.715184i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 18.5iT \) |
| good | 3 | \( 1 + 3.80iT - 81T^{2} \) |
| 5 | \( 1 + 10.4T + 625T^{2} \) |
| 11 | \( 1 + 224. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 312.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 321.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 199. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 450. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 807.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 141. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 667.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.67e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 3.00e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 201. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.51e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 4.97e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.62e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.77e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 3.85e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.62e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.40e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.21e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.84e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.06e3T + 8.85e7T^{2} \) |
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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
−12.45272610316513781988604388098, −11.58538925529230362180370918143, −10.43649020576276530489142012152, −9.146258978029341180989047083226, −8.006033775072084827022464724119, −6.96791830603870769457776778550, −5.63483464304786451761402996650, −4.08291658804494499147618437231, −2.35900143714427768655536510557, −0.30784787579071685226741452808,
2.11140163909709010462742278551, 4.17318593673444129606008345074, 4.89480965811893447690303321703, 7.06311605295124262215761613841, 7.52737935553255687289016754071, 9.495407235836734815232667144894, 9.905407909119638717354661552254, 11.24462787423654551284789224536, 12.39182917145638971239852698821, 13.12674702844154079651406853395
