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
−10.74751851516348895274916710463, −9.977644353667564942994903028408, −9.655642918911110658950249694658, −8.120493280628123772465102841335, −7.78443195766134877916307062056, −7.10638378020444699916025824362, −5.54925987228857641621037879906, −4.76415608265862526512282899119, −3.63148447814922944736768351310, −2.13664982978542386052119142099,
0.24818660102983389173056459610, 1.74159768083169492267982282735, 2.90377281254524027804755672220, 4.70675460615525886918660381067, 5.36358417127120958313713248764, 6.65467965953224870388766036356, 7.50734929911587057256897798790, 8.500110524836803611908346184261, 9.367776953069247841344216201596, 10.08531556365118099343390490084