| L(s) = 1 | + (−0.536 + 1.30i)2-s + (1.62 + 0.594i)3-s + (−1.42 − 1.40i)4-s + (−1.11 − 1.27i)5-s + (−1.65 + 1.80i)6-s + (−2.38 − 0.314i)7-s + (2.60 − 1.10i)8-s + (2.29 + 1.93i)9-s + (2.27 − 0.779i)10-s + (−2.80 − 5.69i)11-s + (−1.48 − 3.13i)12-s + (−0.870 − 2.56i)13-s + (1.69 − 2.95i)14-s + (−1.06 − 2.74i)15-s + (0.0537 + 3.99i)16-s + (2.76 + 2.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.379 + 0.925i)2-s + (0.939 + 0.343i)3-s + (−0.711 − 0.702i)4-s + (−0.500 − 0.571i)5-s + (−0.674 + 0.738i)6-s + (−0.901 − 0.118i)7-s + (0.919 − 0.391i)8-s + (0.764 + 0.645i)9-s + (0.718 − 0.246i)10-s + (−0.846 − 1.71i)11-s + (−0.427 − 0.904i)12-s + (−0.241 − 0.711i)13-s + (0.452 − 0.789i)14-s + (−0.274 − 0.708i)15-s + (0.0134 + 0.999i)16-s + (0.670 + 0.670i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.919056 - 0.331213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.919056 - 0.331213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.536 - 1.30i)T \) |
| 3 | \( 1 + (-1.62 - 0.594i)T \) |
| good | 5 | \( 1 + (1.11 + 1.27i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (2.38 + 0.314i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (2.80 + 5.69i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.870 + 2.56i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-2.76 - 2.76i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.19 + 5.98i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.265 - 2.01i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (7.63 - 0.500i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-6.62 + 3.82i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.71 + 8.61i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.539 - 4.10i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-5.33 + 2.63i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (4.40 - 1.18i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.39 - 3.58i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (5.68 + 6.48i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.678 - 10.3i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-11.2 - 5.55i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-1.14 - 2.76i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.73 - 9.00i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (3.75 + 13.9i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-2.32 - 2.03i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-14.7 + 6.12i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (7.37 + 4.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.32044789455088412725352060592, −9.478711806059963481315433091240, −8.721808392937432166466455282108, −8.032462349520156976210408973036, −7.40691049817635515127963568554, −6.04899667122767084551321903445, −5.18559691416073242126027766101, −3.94740839156662367908084621672, −2.95125634812809960640285294547, −0.56313793670037373973767696280,
1.82420225578305853958824217730, 2.89463766624719013515594220166, 3.68168400906472501382783488982, 4.84486369047569714562084073688, 6.75376281347704253796473693865, 7.51172078481705243570801222152, 8.110261525914140062572723026616, 9.566570842954933147428513930370, 9.655421510754054741106056379478, 10.58825640936182301306989036090
