| L(s) = 1 | − 6.76·2-s + (25.9 − 44.8i)3-s − 82.2·4-s + (−60.8 + 105. i)5-s + (−175. + 303. i)6-s + (496. + 860. i)7-s + 1.42e3·8-s + (−249. − 432. i)9-s + (411. − 712. i)10-s + 3.26e3·11-s + (−2.13e3 + 3.69e3i)12-s + (−5.36e3 − 9.30e3i)13-s + (−3.35e3 − 5.81e3i)14-s + (3.15e3 + 5.46e3i)15-s + 900.·16-s + (−230. − 399. i)17-s + ⋯ |
| L(s) = 1 | − 0.598·2-s + (0.554 − 0.959i)3-s − 0.642·4-s + (−0.217 + 0.376i)5-s + (−0.331 + 0.573i)6-s + (0.547 + 0.947i)7-s + 0.982·8-s + (−0.114 − 0.197i)9-s + (0.130 − 0.225i)10-s + 0.739·11-s + (−0.355 + 0.616i)12-s + (−0.677 − 1.17i)13-s + (−0.327 − 0.566i)14-s + (0.241 + 0.417i)15-s + 0.0549·16-s + (−0.0113 − 0.0197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.35370 - 0.445463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35370 - 0.445463i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.18e5 + 5.54e4i)T \) |
| good | 2 | \( 1 + 6.76T + 128T^{2} \) |
| 3 | \( 1 + (-25.9 + 44.8i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (60.8 - 105. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-496. - 860. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 3.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (5.36e3 + 9.30e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (230. + 399. i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.13e4 + 1.96e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.71e4 + 8.16e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-9.24e4 - 1.60e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.26e5 + 2.19e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.03e5 - 3.52e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.44e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 9.14e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.62e5 + 2.81e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 - 4.16e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (3.41e4 + 5.91e4i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.47e5 - 1.29e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.87e6 + 3.25e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-6.88e5 - 1.19e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.53e6 - 2.66e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.98e5 + 8.63e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (4.24e6 - 7.35e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.03e7T + 8.07e13T^{2} \) |
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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
−14.31989320192636544215180918749, −13.15458761614554230493215760666, −12.13856347156112145302561793356, −10.55814479713906760002292398082, −9.007765430411223128577522008758, −8.175461264604882524700001766471, −7.04578735945939199774761309626, −4.98682508286170268755531421369, −2.67853024414352393088374326893, −0.976668521804811463267301212007,
1.11203895866694688114913508018, 3.92600160354270251968416885846, 4.68087597690700975378804782917, 7.31597627911337285590464697651, 8.693511240337366539634562907923, 9.515253656947101125907137864833, 10.49356170244517911621135381342, 12.05525583544080580818752106988, 13.88777103139867786950298058442, 14.36103856442884443831963392699
