| L(s) = 1 | + (−1.30 + 0.555i)2-s + (−1.67 − 0.423i)3-s + (1.38 − 1.44i)4-s + (3.35 − 1.79i)5-s + (2.41 − 0.382i)6-s + (3.20 − 0.637i)7-s + (−0.995 + 2.64i)8-s + (2.64 + 1.42i)9-s + (−3.36 + 4.19i)10-s + (−2.82 − 2.31i)11-s + (−2.93 + 1.84i)12-s + (−3.67 − 1.96i)13-s + (−3.81 + 2.60i)14-s + (−6.39 + 1.59i)15-s + (−0.175 − 3.99i)16-s + (4.87 + 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.919 + 0.392i)2-s + (−0.969 − 0.244i)3-s + (0.691 − 0.722i)4-s + (1.49 − 0.801i)5-s + (0.987 − 0.156i)6-s + (1.21 − 0.240i)7-s + (−0.352 + 0.935i)8-s + (0.880 + 0.473i)9-s + (−1.06 + 1.32i)10-s + (−0.850 − 0.698i)11-s + (−0.846 + 0.531i)12-s + (−1.01 − 0.544i)13-s + (−1.01 + 0.697i)14-s + (−1.65 + 0.411i)15-s + (−0.0438 − 0.999i)16-s + (1.18 + 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.845115 - 0.361382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.845115 - 0.361382i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 0.555i)T \) |
| 3 | \( 1 + (1.67 + 0.423i)T \) |
| good | 5 | \( 1 + (-3.35 + 1.79i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-3.20 + 0.637i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (2.82 + 2.31i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (3.67 + 1.96i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-4.87 - 2.01i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 3.70i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (0.487 - 0.326i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-4.48 + 3.68i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (4.98 + 4.98i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.84 - 9.37i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-6.15 + 4.11i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.0174 - 0.177i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (0.894 + 0.370i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (4.98 + 4.08i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-0.788 + 0.421i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 13.8i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-5.73 - 0.565i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-3.74 + 0.745i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.19 - 6.01i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.130 - 0.314i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.53 - 5.07i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (2.05 - 3.07i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (2.84 - 2.84i)T - 97iT^{2} \) |
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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
−10.93719120553437517632053867343, −10.15781170725806025059032285374, −9.681948327109631428306947848830, −8.163614732772436636099863033244, −7.77881897333296430098676035430, −6.28471350587836255359723957673, −5.41376928012505586487992027247, −5.09532990904885623585203167423, −2.09755472281896885934566476638, −1.02096498775639029870354023161,
1.63779962818745576863601492028, 2.70252288284928672257506811729, 4.84262216791950037303338164701, 5.64568198236059257609486066662, 6.96577069579948395952919938023, 7.49762697421205146329773563387, 9.122540239198260065735647941735, 9.841603977704052062944515929918, 10.50449734887894819748695849145, 11.12725506318468339309655329605
