| L(s) = 1 | + (−1.60 + 3.32i)2-s + (−14.5 + 21.3i)3-s + (31.4 + 39.3i)4-s + (42.5 + 45.8i)5-s + (−47.6 − 82.5i)6-s + (440. + 254. i)7-s + (−411. + 93.9i)8-s + (22.5 + 57.5i)9-s + (−220. + 68.0i)10-s + (409. − 512. i)11-s + (−1.29e3 + 97.2i)12-s + (−2.19e3 − 677. i)13-s + (−1.55e3 + 1.05e3i)14-s + (−1.59e3 + 240. i)15-s + (−370. + 1.62e3i)16-s + (2.21e3 + 2.05e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.200 + 0.415i)2-s + (−0.538 + 0.790i)3-s + (0.490 + 0.615i)4-s + (0.340 + 0.367i)5-s + (−0.220 − 0.382i)6-s + (1.28 + 0.742i)7-s + (−0.803 + 0.183i)8-s + (0.0309 + 0.0789i)9-s + (−0.220 + 0.0680i)10-s + (0.307 − 0.385i)11-s + (−0.750 + 0.0562i)12-s + (−0.999 − 0.308i)13-s + (−0.565 + 0.385i)14-s + (−0.473 + 0.0713i)15-s + (−0.0905 + 0.396i)16-s + (0.450 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.266328 + 1.52094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266328 + 1.52094i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.95e4 - 3.86e4i)T \) |
| good | 2 | \( 1 + (1.60 - 3.32i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (14.5 - 21.3i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (-42.5 - 45.8i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-440. - 254. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-409. + 512. i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (2.19e3 + 677. i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-2.21e3 - 2.05e3i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (9.95e3 + 3.90e3i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-1.24e4 - 1.87e3i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-1.77e4 - 2.60e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (1.36e3 + 1.82e4i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-5.44e4 + 3.14e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-3.79e4 - 1.82e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (3.11e4 + 3.91e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (8.90e4 - 2.74e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (5.86e4 - 2.57e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.82e5 + 1.36e4i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (1.33e5 - 3.39e5i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-2.61e4 - 1.73e5i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.08e5 + 3.51e5i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-1.91e5 + 3.30e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-7.80e5 - 5.32e5i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (7.71e3 - 1.13e4i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-8.46e5 + 1.06e6i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−15.19064993473270801851979650339, −14.61498593889363362024676908236, −12.62502898525253792994620202648, −11.41792231827215431038072700057, −10.61129596811636615352566701779, −8.909567475181707318005646500478, −7.68645153184436545365279295699, −6.04979039551822418475856987248, −4.67598255342910900256179983432, −2.45020466006612870573086075097,
0.853536217864648311442238002451, 1.90020389337958630659519433207, 4.81763606501312919887683150809, 6.39008120861509168074292506005, 7.57330013340711947893893642041, 9.447269366702205349431543341354, 10.74135475949954133636474930885, 11.74081777022750195229917847391, 12.67016031514465293911576494707, 14.24641474985156254688661925347
