| L(s) = 1 | + (−4.74 − 20.7i)2-s + (8.25 + 7.65i)3-s + (−294. + 141. i)4-s + (467. − 70.4i)5-s + (119. − 207. i)6-s + (375. + 650. i)7-s + (2.63e3 + 3.30e3i)8-s + (−153. − 2.05e3i)9-s + (−3.68e3 − 9.38e3i)10-s + (7.02e3 + 3.38e3i)11-s + (−3.51e3 − 1.08e3i)12-s + (2.11e3 − 5.39e3i)13-s + (1.17e4 − 1.08e4i)14-s + (4.39e3 + 2.99e3i)15-s + (3.01e4 − 3.78e4i)16-s + (−3.46e3 − 521. i)17-s + ⋯ |
| L(s) = 1 | + (−0.419 − 1.83i)2-s + (0.176 + 0.163i)3-s + (−2.29 + 1.10i)4-s + (1.67 − 0.252i)5-s + (0.226 − 0.392i)6-s + (0.413 + 0.716i)7-s + (1.82 + 2.28i)8-s + (−0.0704 − 0.939i)9-s + (−1.16 − 2.96i)10-s + (1.59 + 0.766i)11-s + (−0.586 − 0.180i)12-s + (0.267 − 0.680i)13-s + (1.14 − 1.06i)14-s + (0.336 + 0.229i)15-s + (1.84 − 2.30i)16-s + (−0.170 − 0.0257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.05571 - 1.66437i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05571 - 1.66437i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.94e5 + 1.64e5i)T \) |
| good | 2 | \( 1 + (4.74 + 20.7i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (-8.25 - 7.65i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-467. + 70.4i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (-375. - 650. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-7.02e3 - 3.38e3i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (-2.11e3 + 5.39e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (3.46e3 + 521. i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (2.25e3 - 3.00e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (5.41e4 - 3.69e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.31e5 + 1.21e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (1.47e5 + 4.55e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-1.85e5 + 3.21e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (9.29e3 + 4.07e4i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (-6.90e5 + 3.32e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-1.95e5 - 4.97e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (8.64e5 - 1.08e6i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (-2.62e3 + 811. i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (1.31e5 - 1.75e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (2.77e5 + 1.89e5i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (1.33e6 - 3.40e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (8.63e5 + 1.49e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.01e6 - 3.72e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (4.97e6 + 4.61e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (-3.17e6 - 1.52e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.74790046837211940726341221344, −12.49199549988340308180770120322, −11.80280147847603581785166201105, −10.16193232018402384990800393462, −9.477289951752813703892175375027, −8.711028301927605592170065256916, −5.82643919265620803070400700705, −3.96331881931605444443346144752, −2.23003352056401476377964129787, −1.25796285113308185395496378889,
1.37067720930578416745546036142, 4.71565087609269513326830738631, 6.16369802563124799158455687555, 6.90189904924887805989905753722, 8.511155186910534371062115032596, 9.412433542343192766238058924502, 10.71178028554621599165298174460, 13.48740859518676951836007496920, 13.98514529552936525577293146747, 14.52253917628510033921358919908
