L(s) = 1 | + (7.98 − 0.407i)2-s + (−33.4 − 33.4i)3-s + (63.6 − 6.51i)4-s + (−93.3 − 93.3i)5-s + (−280. − 253. i)6-s + 261.·7-s + (506. − 77.9i)8-s + 1.50e3i·9-s + (−783. − 707. i)10-s + (1.37e3 − 1.37e3i)11-s + (−2.34e3 − 1.90e3i)12-s + (−1.49e3 + 1.49e3i)13-s + (2.08e3 − 106. i)14-s + 6.23e3i·15-s + (4.01e3 − 829. i)16-s + 1.84e3·17-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0509i)2-s + (−1.23 − 1.23i)3-s + (0.994 − 0.101i)4-s + (−0.746 − 0.746i)5-s + (−1.29 − 1.17i)6-s + 0.761·7-s + (0.988 − 0.152i)8-s + 2.06i·9-s + (−0.783 − 0.707i)10-s + (1.03 − 1.03i)11-s + (−1.35 − 1.10i)12-s + (−0.678 + 0.678i)13-s + (0.760 − 0.0387i)14-s + 1.84i·15-s + (0.979 − 0.202i)16-s + 0.375·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.08449 - 1.31135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08449 - 1.31135i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.98 + 0.407i)T \) |
good | 3 | \( 1 + (33.4 + 33.4i)T + 729iT^{2} \) |
| 5 | \( 1 + (93.3 + 93.3i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 261.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.37e3 + 1.37e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.49e3 - 1.49e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 1.84e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-139. - 139. i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 4.37e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (3.65e3 - 3.65e3i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 1.83e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-4.81e4 - 4.81e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 2.54e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (8.60e4 - 8.60e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.48e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-7.99e4 - 7.99e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-2.02e5 + 2.02e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (6.90e3 - 6.90e3i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (2.81e5 + 2.81e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 1.38e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.81e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.59e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (1.95e5 + 1.95e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + 1.10e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.65e5T + 8.32e11T^{2} \) |
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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
−17.06542815641123236627809003975, −16.39693667325039386636184337646, −14.39880228128784473241324128206, −12.99333874171326787462713299031, −11.78332431124346481056383474409, −11.41550598882033340769917791019, −7.83593632503456052565388039032, −6.31890607838481437001110677338, −4.74598429258803943405460784656, −1.19826278157570627641269874128,
3.92276577852914341595327274642, 5.22013145983549651161515164178, 7.06822668478478936554141408887, 10.22295109759800744659224619473, 11.36896278183733800530432421266, 12.14915299152943283588299386726, 14.79779378426987042947964651269, 15.14160132426790788612651659086, 16.59745126623085876074208199599, 17.66666156381241656445602895646