| L(s) = 1 | + (1.49 + 0.973i)2-s + (−0.335 − 0.128i)3-s + (0.487 + 1.09i)4-s + (−0.378 − 0.520i)6-s + (−2.62 + 0.314i)7-s + (0.224 − 1.41i)8-s + (−2.13 − 1.92i)9-s + (−2.82 − 3.13i)11-s + (−0.0225 − 0.430i)12-s + (0.268 − 0.527i)13-s + (−4.24 − 2.08i)14-s + (3.31 − 3.68i)16-s + (0.675 + 0.546i)17-s + (−1.32 − 4.95i)18-s + (−5.83 − 2.59i)19-s + ⋯ |
| L(s) = 1 | + (1.06 + 0.688i)2-s + (−0.193 − 0.0744i)3-s + (0.243 + 0.547i)4-s + (−0.154 − 0.212i)6-s + (−0.992 + 0.119i)7-s + (0.0793 − 0.500i)8-s + (−0.711 − 0.640i)9-s + (−0.851 − 0.946i)11-s + (−0.00651 − 0.124i)12-s + (0.0745 − 0.146i)13-s + (−1.13 − 0.557i)14-s + (0.829 − 0.921i)16-s + (0.163 + 0.132i)17-s + (−0.313 − 1.16i)18-s + (−1.33 − 0.595i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00666 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00666 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.837812 - 0.843419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.837812 - 0.843419i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.314i)T \) |
| good | 2 | \( 1 + (-1.49 - 0.973i)T + (0.813 + 1.82i)T^{2} \) |
| 3 | \( 1 + (0.335 + 0.128i)T + (2.22 + 2.00i)T^{2} \) |
| 11 | \( 1 + (2.82 + 3.13i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.268 + 0.527i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.675 - 0.546i)T + (3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (5.83 + 2.59i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.882 - 1.35i)T + (-9.35 - 21.0i)T^{2} \) |
| 29 | \( 1 + (0.996 - 1.37i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.94 + 0.940i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (4.35 - 0.228i)T + (36.7 - 3.86i)T^{2} \) |
| 41 | \( 1 + (-5.70 + 1.85i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.73 - 3.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.06 + 7.49i)T + (-9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (3.03 - 7.91i)T + (-39.3 - 35.4i)T^{2} \) |
| 59 | \( 1 + (-6.62 - 1.40i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.256 - 1.20i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-7.21 + 8.90i)T + (-13.9 - 65.5i)T^{2} \) |
| 71 | \( 1 + (11.3 + 8.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.182 + 3.48i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (14.0 + 1.47i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-3.01 - 0.477i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-16.6 + 3.54i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (11.5 - 1.83i)T + (92.2 - 29.9i)T^{2} \) |
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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
−10.00174127208155392527827128734, −8.977015788392155922610076324149, −8.153852342291104766129811634501, −6.94668244198620707377981760421, −6.19337828412629254862857219957, −5.78956677746099862077104616729, −4.74460369555812607172778907416, −3.56886464468285675443295580503, −2.84185785660305254682832388936, −0.38316596832801264368701487641,
2.16825653815431523848893367381, 2.91543944425381591957964459640, 4.07472345925632970934433199412, 4.86610139536820324345629685192, 5.76700912322976783468530436426, 6.63687927454210699590305638706, 7.892581581787726251627947597822, 8.600879620245774424867345666500, 10.01308520810792898450221471542, 10.37947154625862252123966295745
