| L(s) = 1 | + (−3.87 − 4.47i)2-s + (13.1 + 8.42i)3-s + (4.12 − 28.6i)4-s + (175. + 80.1i)5-s + (−13.1 − 91.3i)6-s + (148. − 504. i)7-s + (−462. + 297. i)8-s + (100. + 221. i)9-s + (−321. − 1.09e3i)10-s + (1.72e3 + 1.49e3i)11-s + (295. − 341. i)12-s + (−1.32e3 + 388. i)13-s + (−2.83e3 + 1.29e3i)14-s + (1.62e3 + 2.52e3i)15-s + (1.34e3 + 395. i)16-s + (3.04e3 − 437. i)17-s + ⋯ |
| L(s) = 1 | + (−0.484 − 0.559i)2-s + (0.485 + 0.312i)3-s + (0.0644 − 0.448i)4-s + (1.40 + 0.641i)5-s + (−0.0607 − 0.422i)6-s + (0.431 − 1.47i)7-s + (−0.904 + 0.581i)8-s + (0.138 + 0.303i)9-s + (−0.321 − 1.09i)10-s + (1.29 + 1.12i)11-s + (0.171 − 0.197i)12-s + (−0.602 + 0.176i)13-s + (−1.03 + 0.470i)14-s + (0.481 + 0.749i)15-s + (0.328 + 0.0964i)16-s + (0.620 − 0.0891i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.09255 - 1.01236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09255 - 1.01236i\) |
| \(L(4)\) |
|
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 | \( 1 + (8.62e3 + 8.58e3i)T \) |
| good | 2 | \( 1 + (3.87 + 4.47i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (-175. - 80.1i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-148. + 504. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-1.72e3 - 1.49e3i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (1.32e3 - 388. i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-3.04e3 + 437. i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-3.56e3 - 512. i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (4.45e3 + 3.10e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-3.35e4 + 2.15e4i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (4.70e3 - 2.14e3i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (-3.80e4 + 8.32e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (1.43e4 - 2.22e4i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 - 9.95e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (8.03e4 - 2.73e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (2.72e5 - 7.99e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (4.16e4 + 6.48e4i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (2.62e5 - 2.27e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-1.19e5 - 1.37e5i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-4.45e4 + 3.09e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (-2.86e4 - 9.76e4i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (3.51e5 - 1.60e5i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (-1.96e5 + 3.05e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (1.19e6 + 5.47e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
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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
−13.96165679826610500491981612016, −11.99821102172897607697055078030, −10.60492976587485220263810988654, −9.944425180763260837195213646504, −9.389965746585667059717634381002, −7.42968562480421254358570534503, −6.15540494425843976017526272127, −4.35530822467164181621122441056, −2.37404936512341139975796113533, −1.25113881552518356925207317836,
1.46906159407281801495903949012, 3.01582487604772539527609017669, 5.47823652717257659142654982554, 6.42574059863703509560763208969, 8.132149642821488509462901916429, 8.982242540379612682679316029197, 9.555408268494002669779916040108, 11.81351698483489562437359945391, 12.52779250837670774711576367428, 13.80549033849412976664436226507
