L(s) = 1 | + (−0.764 − 0.477i)2-s + (−1.97 + 0.566i)3-s + (−0.520 − 1.06i)4-s + (0.973 + 2.01i)5-s + (1.78 + 0.511i)6-s + (−2.15 + 3.72i)7-s + (−0.300 + 2.85i)8-s + (1.04 − 0.652i)9-s + (0.217 − 2.00i)10-s + (3.98 − 4.42i)11-s + (1.63 + 1.81i)12-s + (−0.0991 − 2.83i)13-s + (3.42 − 1.82i)14-s + (−3.06 − 3.42i)15-s + (0.134 − 0.172i)16-s + (1.20 − 0.0845i)17-s + ⋯ |
L(s) = 1 | + (−0.540 − 0.337i)2-s + (−1.14 + 0.327i)3-s + (−0.260 − 0.533i)4-s + (0.435 + 0.900i)5-s + (0.728 + 0.208i)6-s + (−0.812 + 1.40i)7-s + (−0.106 + 1.01i)8-s + (0.347 − 0.217i)9-s + (0.0688 − 0.634i)10-s + (1.20 − 1.33i)11-s + (0.471 + 0.523i)12-s + (−0.0274 − 0.787i)13-s + (0.915 − 0.486i)14-s + (−0.791 − 0.885i)15-s + (0.0337 − 0.0431i)16-s + (0.293 − 0.0205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0241778 - 0.0864106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0241778 - 0.0864106i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.973 - 2.01i)T \) |
| 19 | \( 1 + (4.35 - 0.223i)T \) |
good | 2 | \( 1 + (0.764 + 0.477i)T + (0.876 + 1.79i)T^{2} \) |
| 3 | \( 1 + (1.97 - 0.566i)T + (2.54 - 1.58i)T^{2} \) |
| 7 | \( 1 + (2.15 - 3.72i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.98 + 4.42i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.0991 + 2.83i)T + (-12.9 + 0.906i)T^{2} \) |
| 17 | \( 1 + (-1.20 + 0.0845i)T + (16.8 - 2.36i)T^{2} \) |
| 23 | \( 1 + (7.63 - 1.07i)T + (22.1 - 6.33i)T^{2} \) |
| 29 | \( 1 + (3.53 + 0.247i)T + (28.7 + 4.03i)T^{2} \) |
| 31 | \( 1 + (2.74 - 1.22i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.43 + 10.5i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.198 + 0.253i)T + (-9.91 - 39.7i)T^{2} \) |
| 43 | \( 1 + (-0.329 - 1.86i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.15 + 0.500i)T + (46.5 + 6.54i)T^{2} \) |
| 53 | \( 1 + (3.98 + 8.16i)T + (-32.6 + 41.7i)T^{2} \) |
| 59 | \( 1 + (1.57 + 3.90i)T + (-42.4 + 40.9i)T^{2} \) |
| 61 | \( 1 + (-7.39 + 1.03i)T + (58.6 - 16.8i)T^{2} \) |
| 67 | \( 1 + (1.91 + 7.67i)T + (-59.1 + 31.4i)T^{2} \) |
| 71 | \( 1 + (4.66 + 4.50i)T + (2.47 + 70.9i)T^{2} \) |
| 73 | \( 1 + (0.00939 - 0.269i)T + (-72.8 - 5.09i)T^{2} \) |
| 79 | \( 1 + (-0.303 + 0.0869i)T + (66.9 - 41.8i)T^{2} \) |
| 83 | \( 1 + (2.92 - 1.30i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.60 - 3.33i)T + (-21.5 + 86.3i)T^{2} \) |
| 97 | \( 1 + (-0.0719 + 0.288i)T + (-85.6 - 45.5i)T^{2} \) |
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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
−10.70244741367308857833905500858, −9.832691957541229971869876282712, −9.187484548651560326100731640344, −8.228204765318473592465098216064, −6.26610546229154656163880303656, −6.05430016551448112864723717667, −5.38892254571655237854289376967, −3.55261784432611098460513389101, −2.16985373101919274874228041030, −0.079280223909802679517950783105,
1.39571296168559823353637248602, 4.00717313439490927898064714902, 4.50081879801746599900027151798, 6.20912147014741952901618754189, 6.71192909180043537062114629416, 7.54708321493769936264455860497, 8.769063226583858580515041606143, 9.713301830789494776464924402328, 10.11826241478817991915477325240, 11.56988295972139121178785416782