| L(s) = 1 | − 46.7i·3-s − 952.·5-s − 3.10e3i·7-s − 2.18e3·9-s + 7.17e3i·11-s + 2.68e4·13-s + 4.45e4i·15-s + 1.46e5·17-s + 2.20e5i·19-s − 1.45e5·21-s + 9.65e4i·23-s + 5.16e5·25-s + 1.02e5i·27-s − 1.69e5·29-s − 4.29e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.52·5-s − 1.29i·7-s − 0.333·9-s + 0.490i·11-s + 0.941·13-s + 0.879i·15-s + 1.75·17-s + 1.68i·19-s − 0.745·21-s + 0.345i·23-s + 1.32·25-s + 0.192i·27-s − 0.239·29-s − 0.465i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.420928516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420928516\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 + 952.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 3.10e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 7.17e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.68e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.46e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.20e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 9.65e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.69e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 4.29e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.94e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.15e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.89e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 8.08e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.21e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 2.47e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 6.21e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.52e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.92e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.17e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.67e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 8.52e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.06e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 5.15e6T + 7.83e15T^{2} \) |
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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
−10.88295350005697545385455065554, −10.04299809267767805318749679753, −8.387753612776677601986395580506, −7.66148068116242400724923851976, −7.10051622085731691432379082998, −5.61833412123836480179537151617, −3.94149243376785453848362239819, −3.56145254969219224565908880826, −1.43937625382632168914530130335, −0.50372469084260486222688005858,
0.815855812692151816641721044719, 2.85201481044860801810832654978, 3.66468671537988042321425621286, 4.91482740875532905739109472199, 5.95547522319621031046056548224, 7.41993208337157933117284538679, 8.504532967604429549880044558968, 9.015536218463112196577883671803, 10.48614848784287023533924417485, 11.49549627404644464232026966155
