| L(s) = 1 | + (4.5 + 7.79i)3-s + (48.1 − 83.4i)5-s + (−40.4 − 123. i)7-s + (−40.5 + 70.1i)9-s + (−171. − 297. i)11-s + 1.17e3·13-s + 866.·15-s + (678. + 1.17e3i)17-s + (−303. + 526. i)19-s + (777. − 869. i)21-s + (1.29e3 − 2.24e3i)23-s + (−3.07e3 − 5.32e3i)25-s − 729·27-s − 3.36e3·29-s + (279. + 484. i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (0.861 − 1.49i)5-s + (−0.312 − 0.950i)7-s + (−0.166 + 0.288i)9-s + (−0.428 − 0.741i)11-s + 1.92·13-s + 0.994·15-s + (0.569 + 0.986i)17-s + (−0.193 + 0.334i)19-s + (0.384 − 0.430i)21-s + (0.511 − 0.886i)23-s + (−0.984 − 1.70i)25-s − 0.192·27-s − 0.743·29-s + (0.0523 + 0.0905i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.580533328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.580533328\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (40.4 + 123. i)T \) |
| good | 5 | \( 1 + (-48.1 + 83.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (171. + 297. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-678. - 1.17e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (303. - 526. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.29e3 + 2.24e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-279. - 484. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-6.98e3 + 1.20e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 8.75e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.14e3 - 3.71e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (5.97e3 + 1.03e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.43e3 - 4.21e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (8.52e3 - 1.47e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.47e3 + 1.12e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (2.84e4 + 4.92e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.58e4 + 2.74e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 5.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.22e4 - 2.12e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 6.91e4T + 8.58e9T^{2} \) |
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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
−10.45026878171884404548416692867, −9.461985552529562994718089433309, −8.622134741624075239359700791302, −8.047576724682088378193008842269, −6.26364591371841206606979642291, −5.57032039036286939342791908531, −4.33213014039590519044817497889, −3.46105003787923893811378997544, −1.60762423547294342757548241310, −0.64867124482323674725382776738,
1.53731781153518036306036827138, 2.64605029632359520653163032311, 3.37245151007128850582961598040, 5.40115398582208412784387161842, 6.27939505087924816948918486016, 6.95997757732398349414750620626, 8.080331528485604164075686230855, 9.270538233600709672503684910030, 9.934546158822749194737076232354, 11.06280138325311138232819656899
