| L(s) = 1 | + (1.71 − 0.392i)2-s + (0.160 + 0.0365i)3-s + (−0.804 + 0.387i)4-s + (5.40 − 4.30i)5-s + 0.289·6-s + 6.65i·7-s + (−6.74 + 5.37i)8-s + (−8.08 − 3.89i)9-s + (7.59 − 9.52i)10-s + (−16.0 − 7.72i)11-s + (−0.143 + 0.0326i)12-s + (12.5 + 15.7i)13-s + (2.60 + 11.4i)14-s + (1.02 − 0.493i)15-s + (−7.25 + 9.09i)16-s + (12.5 − 15.7i)17-s + ⋯ |
| L(s) = 1 | + (0.859 − 0.196i)2-s + (0.0534 + 0.0121i)3-s + (−0.201 + 0.0968i)4-s + (1.08 − 0.861i)5-s + 0.0483·6-s + 0.950i·7-s + (−0.842 + 0.672i)8-s + (−0.898 − 0.432i)9-s + (0.759 − 0.952i)10-s + (−1.45 − 0.701i)11-s + (−0.0119 + 0.00272i)12-s + (0.966 + 1.21i)13-s + (0.186 + 0.816i)14-s + (0.0682 − 0.0328i)15-s + (−0.453 + 0.568i)16-s + (0.737 − 0.925i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.52879 - 0.168724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52879 - 0.168724i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (38.1 - 19.8i)T \) |
| good | 2 | \( 1 + (-1.71 + 0.392i)T + (3.60 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.160 - 0.0365i)T + (8.10 + 3.90i)T^{2} \) |
| 5 | \( 1 + (-5.40 + 4.30i)T + (5.56 - 24.3i)T^{2} \) |
| 7 | \( 1 - 6.65iT - 49T^{2} \) |
| 11 | \( 1 + (16.0 + 7.72i)T + (75.4 + 94.6i)T^{2} \) |
| 13 | \( 1 + (-12.5 - 15.7i)T + (-37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (-12.5 + 15.7i)T + (-64.3 - 281. i)T^{2} \) |
| 19 | \( 1 + (4.58 + 9.52i)T + (-225. + 282. i)T^{2} \) |
| 23 | \( 1 + (-25.0 - 12.0i)T + (329. + 413. i)T^{2} \) |
| 29 | \( 1 + (-8.39 + 1.91i)T + (757. - 364. i)T^{2} \) |
| 31 | \( 1 + (-3.83 - 16.7i)T + (-865. + 416. i)T^{2} \) |
| 37 | \( 1 + 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (7.59 + 33.2i)T + (-1.51e3 + 729. i)T^{2} \) |
| 47 | \( 1 + (-24.4 + 11.7i)T + (1.37e3 - 1.72e3i)T^{2} \) |
| 53 | \( 1 + (54.8 - 68.8i)T + (-625. - 2.73e3i)T^{2} \) |
| 59 | \( 1 + (19.1 - 23.9i)T + (-774. - 3.39e3i)T^{2} \) |
| 61 | \( 1 + (12.5 + 2.87i)T + (3.35e3 + 1.61e3i)T^{2} \) |
| 67 | \( 1 + (17.6 - 8.52i)T + (2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (-22.2 - 46.3i)T + (-3.14e3 + 3.94e3i)T^{2} \) |
| 73 | \( 1 + (-6.25 + 4.98i)T + (1.18e3 - 5.19e3i)T^{2} \) |
| 79 | \( 1 + 8.15T + 6.24e3T^{2} \) |
| 83 | \( 1 + (4.73 - 20.7i)T + (-6.20e3 - 2.98e3i)T^{2} \) |
| 89 | \( 1 + (62.1 + 14.1i)T + (7.13e3 + 3.43e3i)T^{2} \) |
| 97 | \( 1 + (53.1 + 25.5i)T + (5.86e3 + 7.35e3i)T^{2} \) |
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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
−15.56325543301196177920206057578, −14.05956605222306505092109935692, −13.49064979857835335621509255592, −12.43435993897309468381783644882, −11.29744980120692405762759997673, −9.187260354993961123585790758541, −8.668722915116445538940644423318, −5.84893188380943117519791826400, −5.15422122900349819058979772525, −2.85269056726510334614950356051,
3.11551489121402295056963107304, 5.22124258405798365643726749109, 6.29108450299329787364718601464, 8.074055414555457887514682476480, 10.12495271123197942471415287969, 10.65900560384455557986246968091, 12.91558668351927015750196559267, 13.48488491242474817720824998889, 14.46009915733764939833567550156, 15.30404112478968939408708286505
