| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (−2.25 − 1.00i)5-s + (−0.809 + 0.587i)6-s + (−1.01 − 2.44i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (1.23 − 2.14i)10-s + (−2.26 − 2.42i)11-s + (−0.5 − 0.866i)12-s + (2.35 + 1.71i)13-s + (2.53 − 0.755i)14-s + (−0.764 − 2.35i)15-s + (0.913 + 0.406i)16-s + (−0.827 − 7.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.386 + 0.429i)3-s + (−0.489 − 0.103i)4-s + (−1.01 − 0.449i)5-s + (−0.330 + 0.239i)6-s + (−0.384 − 0.923i)7-s + (0.109 − 0.336i)8-s + (−0.0348 + 0.331i)9-s + (0.391 − 0.677i)10-s + (−0.682 − 0.730i)11-s + (−0.144 − 0.249i)12-s + (0.653 + 0.474i)13-s + (0.677 − 0.201i)14-s + (−0.197 − 0.607i)15-s + (0.228 + 0.101i)16-s + (−0.200 − 1.90i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.636622 - 0.387334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636622 - 0.387334i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.01 + 2.44i)T \) |
| 11 | \( 1 + (2.26 + 2.42i)T \) |
| good | 5 | \( 1 + (2.25 + 1.00i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-2.35 - 1.71i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.827 + 7.86i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.188 + 0.0400i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (1.18 + 2.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.615 + 1.89i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.577 + 0.257i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.33 + 1.48i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.79 - 5.52i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 + (-10.1 + 2.16i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (12.0 - 5.37i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (11.2 + 2.38i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-5.94 - 2.64i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-3.46 + 6.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.70 + 1.23i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-13.5 - 2.87i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-0.913 + 8.69i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (2.36 - 1.71i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (6.29 + 10.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.26 + 3.82i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.87352516515957403504388434074, −9.796612371500597117850812666049, −8.962180535574483512751590539818, −8.056372743877797520162160170601, −7.44392229485080712067592888009, −6.35968687753117393846858389309, −4.96085099053107357013838593308, −4.17886082196568525627221016833, −3.14064047897364164557501600035, −0.44831521709235881432088570239,
1.91022963105601218317223540254, 3.14795021173329290010402077740, 3.97857004653403438878446015200, 5.51089863487519154684082045614, 6.66269668525882900570822432252, 7.915905083669326375269232694100, 8.347951211090948763481541959398, 9.439786254084332654529117145358, 10.47870616163134227058989810929, 11.18951612120400004999397645229
