| 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
−11.12725506318468339309655329605, −10.50449734887894819748695849145, −9.841603977704052062944515929918, −9.122540239198260065735647941735, −7.49762697421205146329773563387, −6.96577069579948395952919938023, −5.64568198236059257609486066662, −4.84262216791950037303338164701, −2.70252288284928672257506811729, −1.63779962818745576863601492028,
1.02096498775639029870354023161, 2.09755472281896885934566476638, 5.09532990904885623585203167423, 5.41376928012505586487992027247, 6.28471350587836255359723957673, 7.77881897333296430098676035430, 8.163614732772436636099863033244, 9.681948327109631428306947848830, 10.15781170725806025059032285374, 10.93719120553437517632053867343
