| L(s) = 1 | + (−1.08 − 2.24i)2-s + (−17.4 − 25.5i)3-s + (155. − 195. i)4-s + (618. − 666. i)5-s + (−38.5 + 66.8i)6-s + (963. − 556. i)7-s + (−1.23e3 − 280. i)8-s + (2.04e3 − 5.21e3i)9-s + (−2.16e3 − 668. i)10-s + (1.65e4 + 2.07e4i)11-s + (−7.70e3 − 577. i)12-s + (−1.48e4 + 4.59e3i)13-s + (−2.29e3 − 1.56e3i)14-s + (−2.78e4 − 4.19e3i)15-s + (−1.35e4 − 5.92e4i)16-s + (−1.14e4 + 1.06e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0676 − 0.140i)2-s + (−0.215 − 0.315i)3-s + (0.608 − 0.762i)4-s + (0.989 − 1.06i)5-s + (−0.0297 + 0.0515i)6-s + (0.401 − 0.231i)7-s + (−0.300 − 0.0685i)8-s + (0.312 − 0.795i)9-s + (−0.216 − 0.0668i)10-s + (1.13 + 1.42i)11-s + (−0.371 − 0.0278i)12-s + (−0.521 + 0.160i)13-s + (−0.0597 − 0.0407i)14-s + (−0.549 − 0.0827i)15-s + (−0.206 − 0.904i)16-s + (−0.136 + 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.39579 - 2.01154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39579 - 2.01154i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.70e5 - 3.35e6i)T \) |
| good | 2 | \( 1 + (1.08 + 2.24i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (17.4 + 25.5i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (-618. + 666. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-963. + 556. i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.65e4 - 2.07e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (1.48e4 - 4.59e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (1.14e4 - 1.06e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (-1.81e5 + 7.11e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (3.83e5 - 5.77e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-2.32e5 + 3.40e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (1.02e4 - 1.37e5i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (7.77e5 + 4.49e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-1.02e6 + 4.93e5i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (2.36e6 - 2.97e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-7.73e5 - 2.38e5i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (-3.25e6 - 1.42e7i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-3.56e6 + 2.67e5i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (-8.98e6 - 2.28e7i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-4.97e6 + 3.29e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (2.17e6 + 7.05e6i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (9.15e6 + 1.58e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.42e7 + 1.65e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-4.69e7 - 6.88e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (9.67e7 + 1.21e8i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.90260648931909254948365432043, −12.42681760737067897174158106444, −11.71532185245458043958118381318, −9.824240761282648790462537828394, −9.408938651603766231854783207871, −7.19226358613979179185945907950, −6.00980287049395354632849222089, −4.63915181227051513423604740780, −1.84712647241046646344450518469, −1.05409777225707411080836753311,
2.01581414132229676752613135178, 3.43239350979061243929064470354, 5.65662648725117881864699443154, 6.82528231820383563596986794841, 8.207522380144907338974154609848, 9.868043715225606944742321194663, 11.02568282336451903763207572067, 11.91147909028043917952398858298, 13.68738294088160076317221525516, 14.42060510714615463013294589000
