| L(s) = 1 | + (0.998 + 0.0523i)2-s + (0.555 + 0.686i)3-s + (0.994 + 0.104i)4-s + (2.14 − 0.642i)5-s + (0.519 + 0.714i)6-s + (0.427 + 2.61i)7-s + (0.987 + 0.156i)8-s + (0.461 − 2.17i)9-s + (2.17 − 0.529i)10-s + (−3.71 + 0.789i)11-s + (0.480 + 0.740i)12-s + (−1.44 − 0.737i)13-s + (0.290 + 2.62i)14-s + (1.63 + 1.11i)15-s + (0.978 + 0.207i)16-s + (−2.38 − 6.21i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.0370i)2-s + (0.320 + 0.396i)3-s + (0.497 + 0.0522i)4-s + (0.957 − 0.287i)5-s + (0.211 + 0.291i)6-s + (0.161 + 0.986i)7-s + (0.349 + 0.0553i)8-s + (0.153 − 0.723i)9-s + (0.686 − 0.167i)10-s + (−1.11 + 0.237i)11-s + (0.138 + 0.213i)12-s + (−0.401 − 0.204i)13-s + (0.0776 + 0.702i)14-s + (0.421 + 0.287i)15-s + (0.244 + 0.0519i)16-s + (−0.578 − 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37227 + 0.479328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37227 + 0.479328i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 - 0.0523i)T \) |
| 5 | \( 1 + (-2.14 + 0.642i)T \) |
| 7 | \( 1 + (-0.427 - 2.61i)T \) |
| good | 3 | \( 1 + (-0.555 - 0.686i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (3.71 - 0.789i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.44 + 0.737i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.38 + 6.21i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.889 - 8.46i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.128 + 2.45i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (3.41 - 4.69i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.15 + 4.84i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (5.50 - 3.57i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-4.80 + 1.56i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.22 + 3.22i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.04 + 1.16i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (6.74 - 5.46i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (0.404 - 0.448i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-8.88 + 8.00i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (1.26 - 0.486i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (10.6 + 7.74i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.723 + 1.11i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-1.89 - 4.26i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.58 + 16.3i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-6.29 - 6.98i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.588 - 3.71i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.86175717274142033411303293706, −10.48531200322558302508010178786, −9.724783509951492564769222750850, −8.922869055985382082121709224648, −7.77646486370257008871566782544, −6.44336485667945651472065919325, −5.47899360245567146460109744783, −4.77604397393338529361178644774, −3.16796375198277516727084060009, −2.12810983331789873564151860407,
1.84868099339633100750364573995, 2.91127964810471982508446928832, 4.50935251033707953430284243503, 5.42698678911136574837848538826, 6.69251242138219954106064540456, 7.40158875609741393253238426759, 8.451628434985618017128155396229, 9.846753397510883342245779891648, 10.70756912201733783483913025227, 11.18580419292840860236946381386
