L(s) = 1 | + (−1.17 + 1.59i)2-s + (0.467 − 0.0883i)3-s + (−0.573 − 1.85i)4-s + (−1.51 − 1.64i)5-s + (−0.409 + 0.850i)6-s + (1.41 − 2.23i)7-s + (−0.104 − 0.0364i)8-s + (−2.58 + 1.01i)9-s + (4.41 − 0.479i)10-s + (2.24 − 5.72i)11-s + (−0.431 − 0.817i)12-s + (−2.37 − 0.267i)13-s + (1.90 + 4.89i)14-s + (−0.852 − 0.634i)15-s + (3.39 − 2.31i)16-s + (−2.24 − 0.0838i)17-s + ⋯ |
L(s) = 1 | + (−0.833 + 1.12i)2-s + (0.269 − 0.0510i)3-s + (−0.286 − 0.929i)4-s + (−0.677 − 0.735i)5-s + (−0.167 + 0.347i)6-s + (0.534 − 0.845i)7-s + (−0.0367 − 0.0128i)8-s + (−0.860 + 0.337i)9-s + (1.39 − 0.151i)10-s + (0.677 − 1.72i)11-s + (−0.124 − 0.235i)12-s + (−0.657 − 0.0741i)13-s + (0.509 + 1.30i)14-s + (−0.220 − 0.163i)15-s + (0.848 − 0.578i)16-s + (−0.543 − 0.0203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.574913 - 0.196864i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.574913 - 0.196864i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.51 + 1.64i)T \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
good | 2 | \( 1 + (1.17 - 1.59i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.467 + 0.0883i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-2.24 + 5.72i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (2.37 + 0.267i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (2.24 + 0.0838i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-1.77 + 3.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.116 - 3.10i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (1.81 - 0.414i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-6.67 + 3.85i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.98 + 1.57i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (0.626 + 1.30i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.96 - 5.60i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (5.99 + 4.42i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (4.60 + 2.43i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.849 + 11.3i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.381 - 1.23i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (3.17 + 0.851i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.34 - 14.6i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.88 + 1.39i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-3.91 - 2.26i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.66 + 14.7i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-4.43 - 11.3i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-6.01 - 6.01i)T + 97iT^{2} \) |
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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
−11.62471807992526474996608998642, −11.20167356430647547556427443564, −9.585443783040970764194996598370, −8.685009143721553594098893964274, −8.131064361224190575895065795804, −7.36316507788016603230483295781, −6.11489922389573702566255939859, −4.93288513264412362792560894512, −3.39559758552446880762157090704, −0.61615562399452748488735250981,
2.04668242364036672859675575627, 3.02660067285236525757567046961, 4.52112332379583026210654860799, 6.27395012439851199978589508736, 7.60857074099275663763958389629, 8.578700924549002573214727864017, 9.430902215249210086180887243529, 10.25454692411286501489262866456, 11.32277503529945803399302178270, 12.02890100250707592860618402480