| L(s) = 1 | + (−9.70 + 11.2i)2-s + (13.1 − 8.42i)3-s + (−22.1 − 154. i)4-s + (28.9 − 13.2i)5-s + (−32.8 + 228. i)6-s + (60.2 + 205. i)7-s + (1.14e3 + 736. i)8-s + (100. − 221. i)9-s + (−132. + 452. i)10-s + (−69.4 + 60.1i)11-s + (−1.59e3 − 1.83e3i)12-s + (−3.13e3 − 920. i)13-s + (−2.88e3 − 1.31e3i)14-s + (268. − 417. i)15-s + (−9.80e3 + 2.87e3i)16-s + (891. + 128. i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 + 1.40i)2-s + (0.485 − 0.312i)3-s + (−0.346 − 2.41i)4-s + (0.231 − 0.105i)5-s + (−0.152 + 1.05i)6-s + (0.175 + 0.598i)7-s + (2.23 + 1.43i)8-s + (0.138 − 0.303i)9-s + (−0.132 + 0.452i)10-s + (−0.0521 + 0.0452i)11-s + (−0.920 − 1.06i)12-s + (−1.42 − 0.419i)13-s + (−1.05 − 0.480i)14-s + (0.0794 − 0.123i)15-s + (−2.39 + 0.702i)16-s + (0.181 + 0.0260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.239030 - 0.203655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239030 - 0.203655i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 + 8.42i)T \) |
| 23 | \( 1 + (506. + 1.21e4i)T \) |
| good | 2 | \( 1 + (9.70 - 11.2i)T + (-9.10 - 63.3i)T^{2} \) |
| 5 | \( 1 + (-28.9 + 13.2i)T + (1.02e4 - 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-60.2 - 205. i)T + (-9.89e4 + 6.36e4i)T^{2} \) |
| 11 | \( 1 + (69.4 - 60.1i)T + (2.52e5 - 1.75e6i)T^{2} \) |
| 13 | \( 1 + (3.13e3 + 920. i)T + (4.06e6 + 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-891. - 128. i)T + (2.31e7 + 6.80e6i)T^{2} \) |
| 19 | \( 1 + (5.50e3 - 790. i)T + (4.51e7 - 1.32e7i)T^{2} \) |
| 29 | \( 1 + (-3.48e3 + 2.42e4i)T + (-5.70e8 - 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-3.40e3 - 2.18e3i)T + (3.68e8 + 8.07e8i)T^{2} \) |
| 37 | \( 1 + (7.70e3 + 3.51e3i)T + (1.68e9 + 1.93e9i)T^{2} \) |
| 41 | \( 1 + (3.48e3 + 7.62e3i)T + (-3.11e9 + 3.58e9i)T^{2} \) |
| 43 | \( 1 + (7.69e4 + 1.19e5i)T + (-2.62e9 + 5.75e9i)T^{2} \) |
| 47 | \( 1 + 1.19e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (1.79e3 + 6.12e3i)T + (-1.86e10 + 1.19e10i)T^{2} \) |
| 59 | \( 1 + (1.17e5 + 3.45e4i)T + (3.54e10 + 2.28e10i)T^{2} \) |
| 61 | \( 1 + (4.37e4 - 6.80e4i)T + (-2.14e10 - 4.68e10i)T^{2} \) |
| 67 | \( 1 + (4.14e5 + 3.58e5i)T + (1.28e10 + 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-2.14e5 + 2.47e5i)T + (-1.82e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (8.87e4 + 6.17e5i)T + (-1.45e11 + 4.26e10i)T^{2} \) |
| 79 | \( 1 + (2.15e5 - 7.34e5i)T + (-2.04e11 - 1.31e11i)T^{2} \) |
| 83 | \( 1 + (-8.09e5 - 3.69e5i)T + (2.14e11 + 2.47e11i)T^{2} \) |
| 89 | \( 1 + (2.61e5 + 4.06e5i)T + (-2.06e11 + 4.52e11i)T^{2} \) |
| 97 | \( 1 + (1.25e6 - 5.73e5i)T + (5.45e11 - 6.29e11i)T^{2} \) |
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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
−13.72384090313081432869484495835, −12.23231528689521105939739723829, −10.36863603927343920571606250330, −9.436138425951543026777976512687, −8.429070743767268415671517219806, −7.54803337123938679358952449151, −6.35541015754007104257835330967, −5.08884669440839892777364155660, −2.06239024599667158834898724713, −0.16690925241957681319695091281,
1.66359798050955299243602216941, 2.95087270717105215518807394393, 4.41223775885059516190434998741, 7.26245899055789980238270480011, 8.331993095837333163555652984775, 9.558651370386123991280198404258, 10.16323095146460105473538107369, 11.24545667910302928294534732625, 12.32841904301495184389132545940, 13.46668202879768065348790404986
