L(s) = 1 | + (−0.855 − 0.621i)2-s + (−0.212 − 0.653i)3-s + (−0.272 − 0.838i)4-s + (−0.224 + 0.691i)6-s + 1.01·7-s + (−0.941 + 2.89i)8-s + (2.04 − 1.48i)9-s + (−4.14 − 3.00i)11-s + (−0.490 + 0.356i)12-s + (−4.92 + 3.57i)13-s + (−0.865 − 0.628i)14-s + (1.17 − 0.857i)16-s + (0.986 − 3.03i)17-s − 2.67·18-s + (1.05 − 3.26i)19-s + ⋯ |
L(s) = 1 | + (−0.604 − 0.439i)2-s + (−0.122 − 0.377i)3-s + (−0.136 − 0.419i)4-s + (−0.0916 + 0.282i)6-s + 0.382·7-s + (−0.332 + 1.02i)8-s + (0.681 − 0.495i)9-s + (−1.24 − 0.907i)11-s + (−0.141 + 0.102i)12-s + (−1.36 + 0.992i)13-s + (−0.231 − 0.168i)14-s + (0.294 − 0.214i)16-s + (0.239 − 0.736i)17-s − 0.629·18-s + (0.243 − 0.748i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.102196 + 0.366027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.102196 + 0.366027i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (0.855 + 0.621i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.212 + 0.653i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + (4.14 + 3.00i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (4.92 - 3.57i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.986 + 3.03i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 3.26i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.36 + 1.71i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.479 + 1.47i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.47 - 7.60i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.80 - 4.94i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.50 + 1.09i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 + (1.48 + 4.56i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.10 - 9.55i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.34 - 1.70i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.86 - 1.35i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.43 + 4.42i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.39 + 7.35i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.481 - 0.350i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.41 - 10.5i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.41 + 13.5i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.17 - 3.75i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.35 + 13.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.08531556365118099343390490084, −9.367776953069247841344216201596, −8.500110524836803611908346184261, −7.50734929911587057256897798790, −6.65467965953224870388766036356, −5.36358417127120958313713248764, −4.70675460615525886918660381067, −2.90377281254524027804755672220, −1.74159768083169492267982282735, −0.24818660102983389173056459610,
2.13664982978542386052119142099, 3.63148447814922944736768351310, 4.76415608265862526512282899119, 5.54925987228857641621037879906, 7.10638378020444699916025824362, 7.78443195766134877916307062056, 8.120493280628123772465102841335, 9.655642918911110658950249694658, 9.977644353667564942994903028408, 10.74751851516348895274916710463