| L(s) = 1 | + (−0.00333 + 1.41i)2-s + (0.524 − 1.65i)3-s + (−1.99 − 0.00944i)4-s + (0.407 − 1.12i)5-s + (2.33 + 0.746i)6-s + (−0.984 − 0.173i)7-s + (0.0200 − 2.82i)8-s + (−2.45 − 1.73i)9-s + (1.58 + 0.580i)10-s + (−4.49 + 1.63i)11-s + (−1.06 + 3.29i)12-s + (−2.88 + 2.42i)13-s + (0.248 − 1.39i)14-s + (−1.63 − 1.26i)15-s + (3.99 + 0.0377i)16-s + (−0.870 + 0.502i)17-s + ⋯ |
| L(s) = 1 | + (−0.00236 + 0.999i)2-s + (0.302 − 0.953i)3-s + (−0.999 − 0.00472i)4-s + (0.182 − 0.501i)5-s + (0.952 + 0.304i)6-s + (−0.372 − 0.0656i)7-s + (0.00708 − 0.999i)8-s + (−0.816 − 0.576i)9-s + (0.500 + 0.183i)10-s + (−1.35 + 0.493i)11-s + (−0.307 + 0.951i)12-s + (−0.800 + 0.671i)13-s + (0.0665 − 0.372i)14-s + (−0.422 − 0.325i)15-s + (0.999 + 0.00944i)16-s + (−0.211 + 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00103046 - 0.00868182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00103046 - 0.00868182i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.00333 - 1.41i)T \) |
| 3 | \( 1 + (-0.524 + 1.65i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (-0.407 + 1.12i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (4.49 - 1.63i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.88 - 2.42i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.870 - 0.502i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.65 + 1.53i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 8.69i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.25 + 2.69i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (9.67 - 1.70i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.69 - 4.67i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.07 + 4.85i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.956 - 2.62i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.24 + 12.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 8.85iT - 53T^{2} \) |
| 59 | \( 1 + (11.2 + 4.10i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.764 - 4.33i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.01 + 7.16i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (6.32 + 10.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.88 + 4.98i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.09 + 6.07i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.807 - 0.677i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.4 - 6.01i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.83 - 2.85i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.489334305525939527484328742691, −9.018580648612143163813000468853, −7.977757352073701154078671472008, −7.33215320477196017688542810100, −6.67546938549320699060969618133, −5.53014522392336974316086573870, −4.86141448987999466608050718326, −3.37748735956735856361556553418, −1.89380387814728160560953262964, −0.00399061716725371056826406901,
2.63331624205211817734342327718, 2.84988987544869432349982735005, 4.22599009080981097184977290787, 5.09295022972101229596385258018, 5.99961752334269849386136388002, 7.60795733611058301868678158732, 8.492718963213981199699512938132, 9.228109626852709574969512186356, 10.22138604899656753100896592590, 10.60964546100280530407346921859
