| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.158 + 1.72i)3-s + (−0.978 − 0.207i)4-s + (−2.94 − 0.625i)5-s + (−1.73 − 0.0230i)6-s + (−2.55 − 1.13i)7-s + (0.309 − 0.951i)8-s + (−2.95 + 0.545i)9-s + (0.929 − 2.85i)10-s + (4.22 − 0.897i)11-s + (0.203 − 1.72i)12-s + (−2.25 + 1.00i)13-s + (1.39 − 2.42i)14-s + (0.613 − 5.17i)15-s + (0.913 + 0.406i)16-s + (−1.49 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.0912 + 0.995i)3-s + (−0.489 − 0.103i)4-s + (−1.31 − 0.279i)5-s + (−0.707 − 0.00941i)6-s + (−0.966 − 0.430i)7-s + (0.109 − 0.336i)8-s + (−0.983 + 0.181i)9-s + (0.293 − 0.904i)10-s + (1.27 − 0.270i)11-s + (0.0588 − 0.496i)12-s + (−0.625 + 0.278i)13-s + (0.373 − 0.647i)14-s + (0.158 − 1.33i)15-s + (0.228 + 0.101i)16-s + (−0.362 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.570191 - 0.0991539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.570191 - 0.0991539i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.158 - 1.72i)T \) |
| 41 | \( 1 + (-3.20 + 5.54i)T \) |
| good | 5 | \( 1 + (2.94 + 0.625i)T + (4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (2.55 + 1.13i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-4.22 + 0.897i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.25 - 1.00i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (1.49 - 4.60i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.21 + 4.51i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.28 + 1.46i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-4.74 + 5.27i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (6.27 + 6.97i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.560 - 1.72i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.170 - 1.62i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-0.0805 + 0.766i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (2.62 + 8.09i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.698 + 0.310i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (13.4 + 5.98i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (6.17 + 1.31i)T + (61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (4.64 - 14.2i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + (-1.54 - 2.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.21 - 7.30i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.1 + 9.55i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.19 + 4.66i)T + (-10.1 - 96.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07604330494753357921374183728, −9.314482311438257497967407721521, −8.752721525587788785723583031232, −7.73040154045471811025008323397, −6.89466752802222734502793486307, −5.94259736152163455475124587889, −4.62614100846480569313197116287, −4.02689654353596091373968027368, −3.21912744875344361413741755787, −0.34676702110837178031825978718,
1.24246038852107258484824480865, 2.99402621540014179112480348624, 3.39920780772405517072083202825, 4.85640705857111755981408566835, 6.17290983684223527134548527677, 7.26392406653388353626130492031, 7.56577159494362611779272426162, 8.976195839277560400097391580658, 9.303808180440945275676742857614, 10.60566714218729421514196246648
