| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.381 − 0.174i)3-s + (−0.959 + 0.281i)4-s + (−1.55 + 1.79i)5-s + (−0.118 + 0.402i)6-s + (0.422 − 2.61i)7-s + (0.415 + 0.909i)8-s + (−1.84 − 2.13i)9-s + (2.00 + 1.28i)10-s + (−2.29 − 0.329i)11-s + (0.414 + 0.0596i)12-s + (−2.64 + 4.11i)13-s + (−2.64 − 0.0465i)14-s + (0.907 − 0.414i)15-s + (0.841 − 0.540i)16-s + (−3.97 − 1.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.220 − 0.100i)3-s + (−0.479 + 0.140i)4-s + (−0.697 + 0.804i)5-s + (−0.0481 + 0.164i)6-s + (0.159 − 0.987i)7-s + (0.146 + 0.321i)8-s + (−0.616 − 0.711i)9-s + (0.633 + 0.407i)10-s + (−0.691 − 0.0994i)11-s + (0.119 + 0.0172i)12-s + (−0.732 + 1.13i)13-s + (−0.706 − 0.0124i)14-s + (0.234 − 0.107i)15-s + (0.210 − 0.135i)16-s + (−0.963 − 0.282i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0375836 + 0.149380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0375836 + 0.149380i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.422 + 2.61i)T \) |
| 23 | \( 1 + (3.86 + 2.83i)T \) |
| good | 3 | \( 1 + (0.381 + 0.174i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (1.55 - 1.79i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (2.29 + 0.329i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.64 - 4.11i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.97 + 1.16i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (2.90 - 0.852i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.135 + 0.0399i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.66 + 1.67i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.15 + 1.86i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.43 + 1.23i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.07 - 3.23i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 7.59iT - 47T^{2} \) |
| 53 | \( 1 + (2.62 + 4.08i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (4.43 - 6.89i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.10 + 8.98i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-7.50 + 1.07i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.983 - 6.83i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (1.40 + 4.77i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.18 + 6.50i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.158 + 0.183i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (4.81 - 10.5i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 12.6i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.15113731320892302087252554108, −10.43861704648403264972451336968, −9.396321565634209732361057104013, −8.246012623091934881441434151818, −7.23375092036747358468141993710, −6.37240478056369493635407447307, −4.63559574549254553500591096573, −3.73557235228454002946338294744, −2.40600997764062561938685142788, −0.10949730940303825173364658115,
2.56736682318915060422096171074, 4.50889385223237289130558267140, 5.23436536280485648613155316673, 6.12779498390229210301535718819, 7.76641186135342978813485537041, 8.201550507628604233992791837465, 9.041479488846811196964800381191, 10.27585798025583678898466534308, 11.25649531659250111714375015272, 12.31745995642795598689257623103
