L(s) = 1 | + (−0.898 − 1.55i)2-s + (−0.613 + 1.06i)4-s + (−0.5 − 0.866i)5-s − 3.41·7-s − 1.38·8-s + (−0.898 + 1.55i)10-s + 4.35·11-s + (−2.49 + 4.31i)13-s + (3.06 + 5.31i)14-s + (2.47 + 4.28i)16-s + (−0.0290 − 0.0503i)17-s + (−1.10 + 4.21i)19-s + 1.22·20-s + (−3.91 − 6.77i)22-s + (0.216 − 0.374i)23-s + ⋯ |
L(s) = 1 | + (−0.635 − 1.10i)2-s + (−0.306 + 0.531i)4-s + (−0.223 − 0.387i)5-s − 1.29·7-s − 0.490·8-s + (−0.284 + 0.491i)10-s + 1.31·11-s + (−0.690 + 1.19i)13-s + (0.819 + 1.41i)14-s + (0.618 + 1.07i)16-s + (−0.00704 − 0.0122i)17-s + (−0.253 + 0.967i)19-s + 0.274·20-s + (−0.834 − 1.44i)22-s + (0.0450 − 0.0780i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.591800 + 0.0162518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.591800 + 0.0162518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (1.10 - 4.21i)T \) |
good | 2 | \( 1 + (0.898 + 1.55i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 + (2.49 - 4.31i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.0290 + 0.0503i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.216 + 0.374i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.74 + 4.75i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.592T + 31T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 + (-0.0818 - 0.141i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.00 - 5.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.54 - 9.60i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.69 - 4.67i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.72 + 2.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.15 - 3.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.39 + 5.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.20 - 7.28i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.64 + 6.30i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + (5.59 - 9.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.47 + 7.74i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.884492681305814359633354365160, −9.546889290770351508288024942606, −8.986723378975103725582654984680, −7.888693424869563504207452315158, −6.56189300561636838772418075812, −6.13241492539721958599381545759, −4.40045393701860841962282202820, −3.60288368889574252253708432160, −2.44929417477002821130608798328, −1.18952729947721035808654563353,
0.39852981449867884139451983067, 2.80439026329447324294715082890, 3.64017669374189402394082481084, 5.16390294392428605596036784719, 6.29381724386648942745011043115, 6.73683663503960833758806165859, 7.45766149151985978506223623995, 8.432329756053513149139103519851, 9.273464768267904948348687218387, 9.795599398182794145951756504133