| L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (2.35 + 1.04i)5-s + (−0.809 + 0.587i)6-s + (−1.43 − 2.22i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (1.29 − 2.23i)10-s + (1.89 − 2.72i)11-s + (0.5 + 0.866i)12-s + (−3.71 − 2.69i)13-s + (−2.36 + 1.19i)14-s + (−0.797 − 2.45i)15-s + (0.913 + 0.406i)16-s + (−0.0531 − 0.505i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (−0.386 − 0.429i)3-s + (−0.489 − 0.103i)4-s + (1.05 + 0.469i)5-s + (−0.330 + 0.239i)6-s + (−0.541 − 0.840i)7-s + (−0.109 + 0.336i)8-s + (−0.0348 + 0.331i)9-s + (0.408 − 0.706i)10-s + (0.570 − 0.821i)11-s + (0.144 + 0.249i)12-s + (−1.03 − 0.748i)13-s + (−0.631 + 0.319i)14-s + (−0.205 − 0.633i)15-s + (0.228 + 0.101i)16-s + (−0.0128 − 0.122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.525404 - 1.13244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.525404 - 1.13244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (0.669 + 0.743i)T \) |
| 7 | \( 1 + (1.43 + 2.22i)T \) |
| 11 | \( 1 + (-1.89 + 2.72i)T \) |
| good | 5 | \( 1 + (-2.35 - 1.04i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (3.71 + 2.69i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0531 + 0.505i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-7.29 + 1.55i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (2.63 + 4.56i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.41 + 4.35i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.85 - 1.27i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (6.29 - 6.99i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-0.269 + 0.828i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 + (4.58 - 0.974i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (4.72 - 2.10i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-5.15 - 1.09i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-9.81 - 4.37i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.02 + 3.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.68 + 7.03i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.56 - 1.82i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (0.859 - 8.17i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-4.44 + 3.22i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.45 - 4.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 8.80i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.65461524402833397846731662890, −9.986042202519416660478560186929, −9.359519667066913414289426374839, −7.920639391302986499670783104814, −6.88788890424069695260012446707, −6.00713151903150270237068326410, −5.04164927750332451118223383654, −3.50536408214800432428353728662, −2.43888620847858845860670025366, −0.813252954677126577077963304235,
1.93589785846111797762502359949, 3.71451129495129332966351747723, 5.15735762620554285914578722254, 5.52688927354501453105529080089, 6.61542376490691094568042509804, 7.51918594745117504082740522775, 9.093747579466820406024549907752, 9.467270184685729101085767765216, 9.974779454566061016510903974327, 11.57262512415839629026137512736
