| L(s) = 1 | − 46.7i·3-s + 711. i·5-s + (−686. − 2.30e3i)7-s − 2.18e3·9-s − 770.·11-s − 4.58e3i·13-s + 3.32e4·15-s + 1.34e5i·17-s − 2.30e5i·19-s + (−1.07e5 + 3.21e4i)21-s + 1.31e5·23-s − 1.15e5·25-s + 1.02e5i·27-s + 8.74e5·29-s − 7.50e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.13i·5-s + (−0.286 − 0.958i)7-s − 0.333·9-s − 0.0526·11-s − 0.160i·13-s + 0.657·15-s + 1.61i·17-s − 1.76i·19-s + (−0.553 + 0.165i)21-s + 0.469·23-s − 0.295·25-s + 0.192i·27-s + 1.23·29-s − 0.0812i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.286i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5416158505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5416158505\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| 7 | \( 1 + (686. + 2.30e3i)T \) |
| good | 5 | \( 1 - 711. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 770.T + 2.14e8T^{2} \) |
| 13 | \( 1 + 4.58e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.34e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.30e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.31e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 8.74e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 7.50e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 9.30e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.17e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.81e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.94e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.02e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 8.37e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 3.76e5iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 1.71e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.54e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 3.05e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 3.80e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 5.80e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 4.57e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.33e8iT - 7.83e15T^{2} \) |
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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
−10.00483897894667705573167754336, −8.685084501272869350890859112171, −7.66682354526468286199123208813, −6.77225510135996739895439740444, −6.34779688881248286286006702483, −4.76940187987330918857318820532, −3.48498002094095845200769482265, −2.64982019554167685523218597693, −1.30232580558661439020332401237, −0.11521014635754406612961051388,
1.14775936040573103187157090373, 2.49716281020132026957012924937, 3.67710605585811390371852846581, 4.96224726421281716278017042718, 5.41030313487852614539898399433, 6.67229451324358516105201658067, 8.120656468715598975459226784000, 8.829661038120365790576269432900, 9.546815656738607061685914086341, 10.38102824537925773417934188452
