L(s) = 1 | + (−0.618 + 0.850i)2-s + (−0.535 + 1.64i)3-s + (0.276 + 0.850i)4-s + (−0.319 + 0.232i)5-s + (−1.07 − 1.47i)6-s + (−2.89 − 0.940i)8-s + (−2.42 − 1.76i)9-s − 0.415i·10-s + (−2.36 − 2.32i)11-s − 1.54·12-s + (−2.11 + 2.91i)13-s + (−0.211 − 0.651i)15-s + (1.14 − 0.830i)16-s + (3.00 − 0.975i)18-s + (−0.285 − 0.207i)20-s + ⋯ |
L(s) = 1 | + (−0.437 + 0.601i)2-s + (−0.309 + 0.951i)3-s + (0.138 + 0.425i)4-s + (−0.143 + 0.103i)5-s + (−0.437 − 0.601i)6-s + (−1.02 − 0.332i)8-s + (−0.809 − 0.587i)9-s − 0.131i·10-s + (−0.714 − 0.700i)11-s − 0.446·12-s + (−0.587 + 0.809i)13-s + (−0.0546 − 0.168i)15-s + (0.285 − 0.207i)16-s + (0.707 − 0.229i)18-s + (−0.0639 − 0.0464i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.190000 - 0.331873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.190000 - 0.331873i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.535 - 1.64i)T \) |
| 11 | \( 1 + (2.36 + 2.32i)T \) |
| 13 | \( 1 + (2.11 - 2.91i)T \) |
good | 2 | \( 1 + (0.618 - 0.850i)T + (-0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (0.319 - 0.232i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-5.66 + 4.11i)T^{2} \) |
| 17 | \( 1 + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (7.44 + 2.41i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.0553iT - 43T^{2} \) |
| 47 | \( 1 + (4.15 - 12.7i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.75 - 11.5i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (8.21 + 11.3i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + (3.98 - 2.89i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (3.26 - 4.49i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.64 - 13.2i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + (-29.9 - 92.2i)T^{2} \) |
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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
−11.59737748242168626632581844938, −10.86447843040927547475104762119, −9.777378170565245454797026299374, −9.030288601836306063895584800793, −8.185598805181343375127464227063, −7.19991029369695974137747822989, −6.17824602383755326924164507885, −5.16586556997294746591170803862, −3.89378144577043844424561701361, −2.84351814717762665259778731406,
0.27178871789880816283930673781, 1.86710860337407126779853490304, 2.88130513655555359591128652026, 4.93508389939457458717824115402, 5.80576777748751759234376143774, 6.88286983210130098149522531929, 7.83718380880950620286942508066, 8.706507770335964647355845698171, 10.00618684574458267547104090846, 10.46935418484614564927380202743