| L(s) = 1 | + 4.85·2-s + (12.5 − 21.7i)3-s − 104.·4-s + (−27.5 + 47.7i)5-s + (60.8 − 105. i)6-s + (312. + 540. i)7-s − 1.12e3·8-s + (779. + 1.35e3i)9-s + (−133. + 231. i)10-s + 4.79e3·11-s + (−1.30e3 + 2.26e3i)12-s + (4.73e3 + 8.20e3i)13-s + (1.51e3 + 2.62e3i)14-s + (690. + 1.19e3i)15-s + 7.88e3·16-s + (−4.30e3 − 7.45e3i)17-s + ⋯ |
| L(s) = 1 | + 0.429·2-s + (0.267 − 0.464i)3-s − 0.815·4-s + (−0.0986 + 0.170i)5-s + (0.115 − 0.199i)6-s + (0.343 + 0.595i)7-s − 0.779·8-s + (0.356 + 0.617i)9-s + (−0.0423 + 0.0733i)10-s + 1.08·11-s + (−0.218 + 0.378i)12-s + (0.598 + 1.03i)13-s + (0.147 + 0.255i)14-s + (0.0528 + 0.0915i)15-s + 0.481·16-s + (−0.212 − 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.74157 + 0.872068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74157 + 0.872068i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.20e4 + 5.18e5i)T \) |
| good | 2 | \( 1 - 4.85T + 128T^{2} \) |
| 3 | \( 1 + (-12.5 + 21.7i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (27.5 - 47.7i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-312. - 540. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 4.79e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.73e3 - 8.20e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (4.30e3 + 7.45e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.16e3 + 3.74e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.78e4 - 6.55e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.36e4 + 4.09e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.34e5 - 2.32e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.21e4 + 3.82e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 3.56e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 1.40e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (2.54e4 - 4.40e4i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + 8.69e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + (8.33e5 + 1.44e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.16e6 + 3.74e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.88e6 - 3.27e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.63e6 - 4.55e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.89e6 - 5.01e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.68e6 - 2.91e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-3.12e6 + 5.40e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 5.16e6T + 8.07e13T^{2} \) |
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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
−14.23562855801815323896997342979, −13.67922677713369814484388948427, −12.41819765650725996342030185877, −11.32900408657293554107140491697, −9.436339266716492462041089516206, −8.490694309622925485236676762117, −6.88122236558406594891034094462, −5.21085792691902941552194276468, −3.73875631678514285170856061976, −1.65053131253475052409807258230,
0.78724555246936435555731162696, 3.62038976364084045494290781727, 4.48157111012734050463262857086, 6.23997061825785913087880648536, 8.215941693872442495640136779812, 9.305618117761045670302792368845, 10.50473374503702641423319602107, 12.15774629050645641602866813578, 13.18948652154192205316480240328, 14.39900056971879694174179155274
