| L(s) = 1 | + (0.965 + 2.00i)2-s + (−22.0 − 32.3i)3-s + (36.8 − 46.1i)4-s + (89.0 − 96.0i)5-s + (43.6 − 75.5i)6-s + (−260. + 150. i)7-s + (267. + 60.9i)8-s + (−294. + 751. i)9-s + (278. + 85.9i)10-s + (−688. − 863. i)11-s + (−2.30e3 − 172. i)12-s + (−194. + 59.9i)13-s + (−552. − 376. i)14-s + (−5.07e3 − 765. i)15-s + (−705. − 3.08e3i)16-s + (−2.89e3 + 2.68e3i)17-s + ⋯ |
| L(s) = 1 | + (0.120 + 0.250i)2-s + (−0.817 − 1.19i)3-s + (0.575 − 0.721i)4-s + (0.712 − 0.768i)5-s + (0.201 − 0.349i)6-s + (−0.758 + 0.437i)7-s + (0.521 + 0.119i)8-s + (−0.404 + 1.03i)9-s + (0.278 + 0.0859i)10-s + (−0.517 − 0.648i)11-s + (−1.33 − 0.100i)12-s + (−0.0885 + 0.0272i)13-s + (−0.201 − 0.137i)14-s + (−1.50 − 0.226i)15-s + (−0.172 − 0.754i)16-s + (−0.589 + 0.546i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.263214 - 1.25088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.263214 - 1.25088i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.67e4 + 4.32e4i)T \) |
| good | 2 | \( 1 + (-0.965 - 2.00i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (22.0 + 32.3i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-89.0 + 96.0i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (260. - 150. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (688. + 863. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (194. - 59.9i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (2.89e3 - 2.68e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-6.01e3 + 2.36e3i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (9.94e3 - 1.49e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (8.25e3 - 1.21e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-1.51e3 + 2.01e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (2.20e4 + 1.27e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-1.10e5 + 5.32e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-1.26e5 + 1.58e5i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-8.39e4 - 2.58e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (3.84e4 + 1.68e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.64e5 - 1.23e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-9.62e4 - 2.45e5i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-5.97e4 + 3.96e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (1.53e5 + 4.99e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (3.13e5 + 5.43e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (4.21e5 - 2.87e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (340. + 499. i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-8.80e5 - 1.10e6i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−13.83712107732843534345309648998, −13.02402911663668373404745288806, −11.99905072062793464117574499719, −10.73076867976400634108411829116, −9.233218628807355799976730426254, −7.36006222926031432536887779378, −6.04948502850950988876805484802, −5.52778954122683748938399047612, −2.05389140312083764157768687676, −0.61022115054036160741542929339,
2.73338693879746506746256180825, 4.26019514176819167642144793150, 6.01674083508771214111288402920, 7.32949320375483878851524024289, 9.689643087610467794553138712648, 10.40637055553300889305200240464, 11.36616021484959668172358051092, 12.67016857867335143691725231578, 14.02463135635929319486115283807, 15.64206735521130408180395968143
