| L(s) = 1 | + 15.5i·3-s − 6·5-s − 200. i·7-s − 243·9-s + 644. i·11-s + 2.65e3·13-s − 93.5i·15-s − 7.20e3·17-s + 3.54e3i·19-s + 3.13e3·21-s − 1.74e4i·23-s − 1.55e4·25-s − 3.78e3i·27-s − 1.15e4·29-s + 8.54e3i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.0480·5-s − 0.585i·7-s − 0.333·9-s + 0.484i·11-s + 1.20·13-s − 0.0277i·15-s − 1.46·17-s + 0.516i·19-s + 0.338·21-s − 1.43i·23-s − 0.997·25-s − 0.192i·27-s − 0.473·29-s + 0.286i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.060798358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060798358\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5iT \) |
| good | 5 | \( 1 + 6T + 1.56e4T^{2} \) |
| 7 | \( 1 + 200. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 644. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.65e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.20e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 3.54e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.74e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.15e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 8.54e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 2.23e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.03e5T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.27e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.60e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.68e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.11e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.60e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.17e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 7.08e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.95e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.56e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 8.91e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 2.51e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 5.17e5T + 8.32e11T^{2} \) |
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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
−10.94095689565675097780972874818, −10.41045307311605682857781797228, −9.184284158197369843124791145034, −8.328406724940101714165745960299, −7.01706311405932109915439544591, −5.94012018158778903955430143532, −4.50258364674187048120288851987, −3.70583935074325361871733953681, −2.02802944509387406725098201226, −0.29885969161834938913028930533,
1.28407803668833243997800856541, 2.60575589446991899657678372333, 4.01522902089987152663476470717, 5.61847692480391319823178238113, 6.41246342334536168031765838371, 7.67105656188679982942401945587, 8.672389780173542911616842460442, 9.479291894298240860542527200316, 11.22037719508825778687115133056, 11.37154169502675660076633119485
