| L(s) = 1 | + (0.415 − 0.909i)3-s + (0.531 − 0.612i)5-s + (0.0574 − 0.0368i)7-s + (−0.654 − 0.755i)9-s + (0.604 − 4.20i)11-s + (−4.84 − 3.11i)13-s + (−0.336 − 0.737i)15-s + (0.227 + 0.0667i)17-s + (4.34 − 1.27i)19-s + (−0.00971 − 0.0675i)21-s + (4.39 + 1.92i)23-s + (0.618 + 4.29i)25-s + (−0.959 + 0.281i)27-s + (1.20 + 0.355i)29-s + (−3.60 − 7.88i)31-s + ⋯ |
| L(s) = 1 | + (0.239 − 0.525i)3-s + (0.237 − 0.274i)5-s + (0.0216 − 0.0139i)7-s + (−0.218 − 0.251i)9-s + (0.182 − 1.26i)11-s + (−1.34 − 0.864i)13-s + (−0.0869 − 0.190i)15-s + (0.0551 + 0.0161i)17-s + (0.997 − 0.292i)19-s + (−0.00211 − 0.0147i)21-s + (0.916 + 0.400i)23-s + (0.123 + 0.859i)25-s + (−0.184 + 0.0542i)27-s + (0.224 + 0.0659i)29-s + (−0.647 − 1.41i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0206 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03046 - 1.05198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03046 - 1.05198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (-4.39 - 1.92i)T \) |
| good | 5 | \( 1 + (-0.531 + 0.612i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.0574 + 0.0368i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.604 + 4.20i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (4.84 + 3.11i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.227 - 0.0667i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.34 + 1.27i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.20 - 0.355i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.60 + 7.88i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (6.18 + 7.14i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.67 + 3.08i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 2.63i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 + (4.88 - 3.13i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.49 - 4.17i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.75 - 8.22i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.11 + 7.75i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.03 - 7.21i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.50 - 0.735i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.66 - 3.64i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.83 - 7.89i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.06 + 13.2i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.91 - 7.98i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.64395242358268405875476533282, −9.450687102793633507042442915313, −8.914717890886839288290625594985, −7.71627977116106383191859711773, −7.20164965200524524525620963350, −5.79648663817544326520303910938, −5.21012931973222592693976261416, −3.57240613381817643897870100698, −2.53240236354651292639134722332, −0.847345122675603767055927003930,
1.96386342174349909922505111283, 3.14853130676407806267033392063, 4.55190704882767587003402559717, 5.13073665406272981687741145728, 6.69951401740915698573977450180, 7.26692971266559270391481184006, 8.467121109600141986755893807924, 9.553907445865000159398738348869, 9.892721678280333191452740974856, 10.84495224949672990949682713115
