L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.04 + 2.43i)7-s − 0.999·8-s + (−1.38 + 0.800i)11-s − 0.770·13-s + (1.58 + 2.11i)14-s + (−0.5 + 0.866i)16-s + (3.05 − 1.76i)17-s + (−3.06 − 1.77i)19-s + 1.60i·22-s + (1.61 − 2.79i)23-s + (−0.385 + 0.667i)26-s + (2.62 − 0.313i)28-s + 0.700i·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.393 + 0.919i)7-s − 0.353·8-s + (−0.417 + 0.241i)11-s − 0.213·13-s + (0.423 + 0.566i)14-s + (−0.125 + 0.216i)16-s + (0.739 − 0.427i)17-s + (−0.703 − 0.406i)19-s + 0.341i·22-s + (0.336 − 0.582i)23-s + (−0.0755 + 0.130i)26-s + (0.496 − 0.0592i)28-s + 0.130i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8996736797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8996736797\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.04 - 2.43i)T \) |
good | 11 | \( 1 + (1.38 - 0.800i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.770T + 13T^{2} \) |
| 17 | \( 1 + (-3.05 + 1.76i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.06 + 1.77i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.61 + 2.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.700iT - 29T^{2} \) |
| 31 | \( 1 + (-1.13 + 0.656i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.792 - 0.457i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 + 9.26iT - 43T^{2} \) |
| 47 | \( 1 + (-2.31 - 1.33i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.64 + 8.04i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.56 - 2.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.43 + 5.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.90 - 3.40i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.47iT - 71T^{2} \) |
| 73 | \( 1 + (5.51 + 9.55i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.45 + 2.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 + (-4.40 + 7.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−8.603268996158138926689625468029, −7.61620316347455717512249935811, −6.72247736904342260661315688726, −5.93898137527570194901438998419, −5.17521046624131173646719342524, −4.52563338016603755938507117436, −3.36518244393732301015046318190, −2.68741268084615041856392735247, −1.82421739628683717022325932798, −0.24804044845110198517723609241,
1.28063258054392483063214672631, 2.80003635823325760521736921956, 3.62259438764084514098501391120, 4.37883393290604206626800618486, 5.23494871956694191165505532562, 6.09589300820120984758442967364, 6.66980857577660957065868781449, 7.64520940062525863669942273164, 7.917155317042534944296854964177, 8.916069446414014585541045637504