| L(s) = 1 | + (−0.0523 − 0.998i)2-s + (−0.366 + 0.955i)3-s + (−0.994 + 0.104i)4-s + (1.54 − 1.61i)5-s + (0.973 + 0.316i)6-s + (−2.56 − 0.645i)7-s + (0.156 + 0.987i)8-s + (1.45 + 1.30i)9-s + (−1.69 − 1.45i)10-s + (2.64 − 2.00i)11-s + (0.265 − 0.989i)12-s + (2.72 + 1.39i)13-s + (−0.510 + 2.59i)14-s + (0.982 + 2.06i)15-s + (0.978 − 0.207i)16-s + (−0.237 + 4.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.0370 − 0.706i)2-s + (−0.211 + 0.551i)3-s + (−0.497 + 0.0522i)4-s + (0.689 − 0.724i)5-s + (0.397 + 0.129i)6-s + (−0.969 − 0.244i)7-s + (0.0553 + 0.349i)8-s + (0.483 + 0.435i)9-s + (−0.536 − 0.460i)10-s + (0.797 − 0.603i)11-s + (0.0765 − 0.285i)12-s + (0.756 + 0.385i)13-s + (−0.136 + 0.693i)14-s + (0.253 + 0.534i)15-s + (0.244 − 0.0519i)16-s + (−0.0575 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26711 - 0.771807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26711 - 0.771807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0523 + 0.998i)T \) |
| 5 | \( 1 + (-1.54 + 1.61i)T \) |
| 7 | \( 1 + (2.56 + 0.645i)T \) |
| 11 | \( 1 + (-2.64 + 2.00i)T \) |
| good | 3 | \( 1 + (0.366 - 0.955i)T + (-2.22 - 2.00i)T^{2} \) |
| 13 | \( 1 + (-2.72 - 1.39i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.237 - 4.52i)T + (-16.9 - 1.77i)T^{2} \) |
| 19 | \( 1 + (-0.380 + 3.62i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.339 + 1.26i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.64 + 3.64i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.82 + 8.57i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-8.61 + 3.30i)T + (27.4 - 24.7i)T^{2} \) |
| 41 | \( 1 + (-2.62 + 3.61i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.98 - 8.98i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.93 + 3.99i)T + (9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (1.60 + 2.46i)T + (-21.5 + 48.4i)T^{2} \) |
| 59 | \( 1 + (0.632 + 6.01i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-2.35 - 11.0i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.120 + 0.451i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.216 + 0.666i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.87 + 3.94i)T + (15.1 + 71.4i)T^{2} \) |
| 79 | \( 1 + (8.69 + 7.83i)T + (8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (1.69 + 3.33i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-2.00 - 3.47i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.88 - 11.5i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.14974729236144031666401941195, −9.355895258230572236155439004372, −8.985082839006308469510704767471, −7.77392602800731926136244521377, −6.30921029628597525121568167199, −5.77306025154376909792162145856, −4.33135627294647983281198715315, −3.92877939039909834579623347071, −2.36847992940384130077973876553, −0.960020318296133820742644962293,
1.29594349609390690545885619616, 2.94572298138814245326215123923, 4.03883392864776355657236205990, 5.56946433958566168709987247001, 6.24444563070997290170130912832, 6.90987584822379523380072496842, 7.46444378769282508553622908254, 8.902236075450481954313437216736, 9.567144968943158782542382734690, 10.16536204473154564188893439415
