| L(s) = 1 | + (−1.99 − 4.13i)2-s + (7.04 + 10.3i)3-s + (−3.17 + 3.97i)4-s + (−25.9 + 28.0i)5-s + (28.7 − 49.7i)6-s + (−65.6 + 37.8i)7-s + (−48.8 − 11.1i)8-s + (−27.5 + 70.3i)9-s + (167. + 51.7i)10-s + (33.3 + 41.8i)11-s + (−63.4 − 4.75i)12-s + (222. − 68.7i)13-s + (287. + 195. i)14-s + (−472. − 71.2i)15-s + (69.3 + 303. i)16-s + (−53.0 + 49.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.498 − 1.03i)2-s + (0.783 + 1.14i)3-s + (−0.198 + 0.248i)4-s + (−1.03 + 1.12i)5-s + (0.797 − 1.38i)6-s + (−1.33 + 0.773i)7-s + (−0.763 − 0.174i)8-s + (−0.340 + 0.868i)9-s + (1.67 + 0.517i)10-s + (0.275 + 0.346i)11-s + (−0.440 − 0.0330i)12-s + (1.31 − 0.406i)13-s + (1.46 + 0.999i)14-s + (−2.10 − 0.316i)15-s + (0.270 + 1.18i)16-s + (−0.183 + 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0349 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.627261 + 0.605724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627261 + 0.605724i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (231. + 1.83e3i)T \) |
| good | 2 | \( 1 + (1.99 + 4.13i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (-7.04 - 10.3i)T + (-29.5 + 75.4i)T^{2} \) |
| 5 | \( 1 + (25.9 - 28.0i)T + (-46.7 - 623. i)T^{2} \) |
| 7 | \( 1 + (65.6 - 37.8i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-33.3 - 41.8i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-222. + 68.7i)T + (2.35e4 - 1.60e4i)T^{2} \) |
| 17 | \( 1 + (53.0 - 49.2i)T + (6.24e3 - 8.32e4i)T^{2} \) |
| 19 | \( 1 + (6.50 - 2.55i)T + (9.55e4 - 8.86e4i)T^{2} \) |
| 23 | \( 1 + (-375. + 56.6i)T + (2.67e5 - 8.24e4i)T^{2} \) |
| 29 | \( 1 + (757. - 1.11e3i)T + (-2.58e5 - 6.58e5i)T^{2} \) |
| 31 | \( 1 + (-8.68 + 115. i)T + (-9.13e5 - 1.37e5i)T^{2} \) |
| 37 | \( 1 + (-1.93e3 - 1.11e3i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (2.36e3 - 1.14e3i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (2.23e3 - 2.80e3i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (-2.17e3 - 670. i)T + (6.51e6 + 4.44e6i)T^{2} \) |
| 59 | \( 1 + (946. + 4.14e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (-677. + 50.7i)T + (1.36e7 - 2.06e6i)T^{2} \) |
| 67 | \( 1 + (-766. - 1.95e3i)T + (-1.47e7 + 1.37e7i)T^{2} \) |
| 71 | \( 1 + (691. - 4.58e3i)T + (-2.42e7 - 7.49e6i)T^{2} \) |
| 73 | \( 1 + (582. + 1.88e3i)T + (-2.34e7 + 1.59e7i)T^{2} \) |
| 79 | \( 1 + (-1.84e3 - 3.19e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-7.31e3 + 4.98e3i)T + (1.73e7 - 4.41e7i)T^{2} \) |
| 89 | \( 1 + (3.63e3 + 5.33e3i)T + (-2.29e7 + 5.84e7i)T^{2} \) |
| 97 | \( 1 + (-7.53e3 - 9.44e3i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−15.33214562468000425182328998033, −14.85676104987586354601572589030, −12.87861327087737265419656620612, −11.47499019552656968463067069387, −10.56714340782342070943715768356, −9.585112658693465524191026554845, −8.613508835328752120128379235521, −6.49244249860434793694367229728, −3.57700246601708293738237193926, −3.02011833248620646058353043335,
0.60537109644070520302895717730, 3.60790025902589033559666603925, 6.37019678762605145055677498833, 7.38790482555799622090153297957, 8.358437511311423336922214754660, 9.191082575981219112544250862378, 11.66210206076362665862002299750, 12.93623628254141715117311445284, 13.56147070601260480036196514859, 15.23340096612110783840456574773
