| L(s) = 1 | + (−0.822 − 2.88i)3-s − 1.24i·5-s + 2.64·7-s + (−7.64 + 4.74i)9-s + 7.01i·11-s + 11.6·13-s + (−3.58 + 1.02i)15-s + 4.52i·17-s − 16.2·19-s + (−2.17 − 7.63i)21-s − 25.5i·23-s + 23.4·25-s + (19.9 + 18.1i)27-s − 9.49i·29-s + 28.7·31-s + ⋯ |
| L(s) = 1 | + (−0.274 − 0.961i)3-s − 0.248i·5-s + 0.377·7-s + (−0.849 + 0.527i)9-s + 0.637i·11-s + 0.895·13-s + (−0.238 + 0.0681i)15-s + 0.266i·17-s − 0.854·19-s + (−0.103 − 0.363i)21-s − 1.11i·23-s + 0.938·25-s + (0.740 + 0.672i)27-s − 0.327i·29-s + 0.926·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.797225087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797225087\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.822 + 2.88i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 1.24iT - 25T^{2} \) |
| 11 | \( 1 - 7.01iT - 121T^{2} \) |
| 13 | \( 1 - 11.6T + 169T^{2} \) |
| 17 | \( 1 - 4.52iT - 289T^{2} \) |
| 19 | \( 1 + 16.2T + 361T^{2} \) |
| 23 | \( 1 + 25.5iT - 529T^{2} \) |
| 29 | \( 1 + 9.49iT - 841T^{2} \) |
| 31 | \( 1 - 28.7T + 961T^{2} \) |
| 37 | \( 1 - 33.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 67.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 15.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 92.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 15.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 70.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 74.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 23.1T + 9.40e3T^{2} \) |
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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
−9.005087073369213552214208823836, −8.317926273182033016422166640010, −7.75032434496438925078121353530, −6.58202274164577097686986392647, −6.25293887218820143154995600103, −5.04231425063808481734025521263, −4.27095739429517601965337149893, −2.79614731523327093838925229113, −1.77815118065169332903092205983, −0.70317908654106436638004781304,
0.947611530417737044759561518104, 2.63683552739105504748806695233, 3.63320970625435457393229937649, 4.41091813377127825630965893297, 5.44731153125577370598158828743, 6.08114760229647795429654170361, 7.05343130713134123118481641626, 8.247515898646818363825156892813, 8.799389751651145172532580359401, 9.591509344728980044211494510101
