| L(s) = 1 | + (0.998 − 0.0627i)2-s + (0.904 − 0.425i)3-s + (0.992 − 0.125i)4-s + (−2.16 − 0.550i)5-s + (0.876 − 0.481i)6-s + (−1.77 − 2.44i)7-s + (0.982 − 0.187i)8-s + (0.637 − 0.770i)9-s + (−2.19 − 0.413i)10-s + (−0.0803 − 1.27i)11-s + (0.844 − 0.535i)12-s + (−3.57 − 2.95i)13-s + (−1.92 − 2.33i)14-s + (−2.19 + 0.424i)15-s + (0.968 − 0.248i)16-s + (0.178 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0443i)2-s + (0.522 − 0.245i)3-s + (0.496 − 0.0626i)4-s + (−0.969 − 0.246i)5-s + (0.357 − 0.196i)6-s + (−0.672 − 0.925i)7-s + (0.347 − 0.0662i)8-s + (0.212 − 0.256i)9-s + (−0.694 − 0.130i)10-s + (−0.0242 − 0.385i)11-s + (0.243 − 0.154i)12-s + (−0.991 − 0.820i)13-s + (−0.515 − 0.623i)14-s + (−0.566 + 0.109i)15-s + (0.242 − 0.0621i)16-s + (0.0432 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04298 - 1.50101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04298 - 1.50101i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 + 0.0627i)T \) |
| 3 | \( 1 + (-0.904 + 0.425i)T \) |
| 5 | \( 1 + (2.16 + 0.550i)T \) |
| good | 7 | \( 1 + (1.77 + 2.44i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (0.0803 + 1.27i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (3.57 + 2.95i)T + (2.43 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.178 + 1.41i)T + (-16.4 - 4.22i)T^{2} \) |
| 19 | \( 1 + (-0.364 + 0.775i)T + (-12.1 - 14.6i)T^{2} \) |
| 23 | \( 1 + (-0.148 - 0.375i)T + (-16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (0.198 - 0.186i)T + (1.82 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-2.61 - 0.330i)T + (30.0 + 7.70i)T^{2} \) |
| 37 | \( 1 + (1.64 + 6.41i)T + (-32.4 + 17.8i)T^{2} \) |
| 41 | \( 1 + (-8.26 - 3.27i)T + (29.8 + 28.0i)T^{2} \) |
| 43 | \( 1 + (6.81 + 2.21i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-7.63 - 1.45i)T + (43.6 + 17.3i)T^{2} \) |
| 53 | \( 1 + (0.370 - 0.674i)T + (-28.3 - 44.7i)T^{2} \) |
| 59 | \( 1 + (-0.894 - 1.40i)T + (-25.1 + 53.3i)T^{2} \) |
| 61 | \( 1 + (-3.27 + 1.29i)T + (44.4 - 41.7i)T^{2} \) |
| 67 | \( 1 + (4.26 - 4.54i)T + (-4.20 - 66.8i)T^{2} \) |
| 71 | \( 1 + (0.102 - 0.539i)T + (-66.0 - 26.1i)T^{2} \) |
| 73 | \( 1 + (-4.99 - 3.17i)T + (31.0 + 66.0i)T^{2} \) |
| 79 | \( 1 + (0.414 + 0.881i)T + (-50.3 + 60.8i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 5.05i)T + (52.9 + 63.9i)T^{2} \) |
| 89 | \( 1 + (4.64 - 7.31i)T + (-37.8 - 80.5i)T^{2} \) |
| 97 | \( 1 + (-1.87 - 1.99i)T + (-6.09 + 96.8i)T^{2} \) |
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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
−10.20897097187115281669208022030, −9.251569079375946718960730399212, −8.104418159657602374573003398265, −7.41631924958194361248820682449, −6.80212807639962738452273725391, −5.47956493957912915439483438875, −4.39393340003033743472572017250, −3.55529116551978946016343177306, −2.70120913913799036518105470297, −0.68091894084915003425666126283,
2.26233720851693986269196785953, 3.16092394481438408336519941643, 4.16382863424276019569961096528, 5.00092715709789729284999723393, 6.25987390418838925561164759484, 7.10156654362458069382446533482, 7.938286843898617277715147217240, 8.911287264921218661968319771651, 9.737155095152008545449755967358, 10.65044141003393795000411572650
