L(s) = 1 | + (13.9 − 13.9i)3-s + (−25 + 50i)5-s + (−125. − 125. i)7-s − 147i·9-s − 418. i·11-s + (−695 − 695i)13-s + (349. + 1.04e3i)15-s + (−95 + 95i)17-s + 837.·19-s − 3.50e3·21-s + (−2.55e3 + 2.55e3i)23-s + (−1.87e3 − 2.50e3i)25-s + (1.34e3 + 1.34e3i)27-s − 2.87e3i·29-s − 9.63e3i·31-s + ⋯ |
L(s) = 1 | + (0.895 − 0.895i)3-s + (−0.447 + 0.894i)5-s + (−0.969 − 0.969i)7-s − 0.604i·9-s − 1.04i·11-s + (−1.14 − 1.14i)13-s + (0.400 + 1.20i)15-s + (−0.0797 + 0.0797i)17-s + 0.532·19-s − 1.73·21-s + (−1.00 + 1.00i)23-s + (−0.600 − 0.800i)25-s + (0.353 + 0.353i)27-s − 0.635i·29-s − 1.80i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.320397 - 1.12784i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.320397 - 1.12784i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (25 - 50i)T \) |
good | 3 | \( 1 + (-13.9 + 13.9i)T - 243iT^{2} \) |
| 7 | \( 1 + (125. + 125. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 418. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (695 + 695i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (95 - 95i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 837.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.55e3 - 2.55e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 2.87e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 9.63e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (5.15e3 - 5.15e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 9.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (1.55e3 - 1.55e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.26e4 - 1.26e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.13e4 - 1.13e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 837.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (1.73e4 + 1.73e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 5.82e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.40e4 + 2.40e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 9.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.56e4 + 3.56e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 7.42e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (4.92e4 - 4.92e4i)T - 8.58e9iT^{2} \) |
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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
−13.33247799395804651747420343119, −12.03752629148004012592455305767, −10.66593693209956360062633442941, −9.639749104703631221728279811795, −7.84940357638755562001592007354, −7.47303353495610373863886479269, −6.12312472288650551408264291420, −3.62918116571160645619895480669, −2.66337866477535571202758743285, −0.42258539250052499854667536783,
2.39083404713299267969880621805, 3.94845914941493872525268872500, 5.06849877393356948645880415044, 6.99572802079066391152478227434, 8.626417923526845962841878586071, 9.302502587072473171265106723440, 10.05079115765791052164212147000, 12.12707990750174831211456762880, 12.45785888934787171269631194944, 14.10975193995883017069999814904