| L(s) = 1 | + 9i·3-s + 46.1i·5-s − 59.9·7-s − 81·9-s − 400. i·11-s − 351. i·13-s − 415.·15-s + 1.66e3·17-s + 564. i·19-s − 539. i·21-s + 3.83e3·23-s + 994.·25-s − 729i·27-s + 5.94e3i·29-s + 2.67e3·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.825i·5-s − 0.462·7-s − 0.333·9-s − 0.996i·11-s − 0.577i·13-s − 0.476·15-s + 1.39·17-s + 0.358i·19-s − 0.266i·21-s + 1.51·23-s + 0.318·25-s − 0.192i·27-s + 1.31i·29-s + 0.500·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.885033432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885033432\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
| good | 5 | \( 1 - 46.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 59.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 400. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 351. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 564. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.83e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.94e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 771. iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 5.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.51e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.10e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.21e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.23e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.58e5T + 8.58e9T^{2} \) |
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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
−10.66084220985412299239092512390, −10.01765857279568384134441360166, −8.993226048478772591054304055731, −8.013979709238368006866670324272, −6.90999429689976172632114379804, −5.93285011654855162791950562102, −4.98611142876199070335693433938, −3.27222813613851451457048225901, −3.11412480465990939999585307500, −0.994199208038070722862142231870,
0.58521615543133923623183863939, 1.66908964185988801303935278232, 3.03525448645135436036981567423, 4.47217593196404323049889326618, 5.39798864787588173408594551577, 6.61529742909922884325935932104, 7.41699048634025506391247499681, 8.446577645654690098068855752097, 9.363404914865957824549809180020, 10.10032173289784457734125797520
