| L(s) = 1 | + 1.13·2-s + (−21.7 + 37.6i)3-s − 126.·4-s + (−195. + 338. i)5-s + (−24.7 + 42.8i)6-s + (−513. − 889. i)7-s − 289.·8-s + (146. + 254. i)9-s + (−222. + 385. i)10-s + 5.63e3·11-s + (2.75e3 − 4.77e3i)12-s + (−2.18e3 − 3.78e3i)13-s + (−584. − 1.01e3i)14-s + (−8.51e3 − 1.47e4i)15-s + 1.58e4·16-s + (−3.41e3 − 5.91e3i)17-s + ⋯ |
| L(s) = 1 | + 0.100·2-s + (−0.465 + 0.805i)3-s − 0.989·4-s + (−0.700 + 1.21i)5-s + (−0.0467 + 0.0810i)6-s + (−0.566 − 0.980i)7-s − 0.200·8-s + (0.0671 + 0.116i)9-s + (−0.0704 + 0.121i)10-s + 1.27·11-s + (0.460 − 0.797i)12-s + (−0.275 − 0.477i)13-s + (−0.0569 − 0.0986i)14-s + (−0.651 − 1.12i)15-s + 0.969·16-s + (−0.168 − 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.275729 - 0.192634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.275729 - 0.192634i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.63e5 - 2.39e5i)T \) |
| good | 2 | \( 1 - 1.13T + 128T^{2} \) |
| 3 | \( 1 + (21.7 - 37.6i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (195. - 338. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (513. + 889. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 5.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (2.18e3 + 3.78e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (3.41e3 + 5.91e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.00e4 - 3.46e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.59e4 + 4.48e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.19e5 + 2.06e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.02e5 + 1.77e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.43e5 - 4.22e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 2.17e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 1.12e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (2.15e4 - 3.72e4i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 - 2.84e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (1.01e6 + 1.75e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.02e6 + 1.76e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (7.59e5 + 1.31e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.53e6 + 2.65e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.27e6 + 3.94e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (9.22e5 - 1.59e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-2.54e6 + 4.41e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 3.29e6T + 8.07e13T^{2} \) |
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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
−14.40157184250260988790209564767, −13.19305063723266187813669584875, −11.60871673679060337711678522357, −10.43115563143495949023326679274, −9.704344020864630803719261122177, −7.82595421966963211173582639153, −6.34918449003218776384246831927, −4.37377762130843803022017622903, −3.60813003182056981406424944363, −0.17941408308908672312742643766,
1.16858021337323954047097215267, 3.97681194991034653915526418372, 5.37699223110789081999795150254, 6.90226736154114710208330648881, 8.823498435573428344000046087057, 9.173128908191824289413097680885, 11.65648864514830765343961190796, 12.52975045893432202365806488482, 12.98720132920108557723772219759, 14.59656705996762627091777079042
