L(s) = 1 | − 14i·3-s + (−55 − 10i)5-s − 158i·7-s + 47·9-s + 148·11-s − 684i·13-s + (−140 + 770i)15-s − 2.04e3i·17-s + 2.22e3·19-s − 2.21e3·21-s − 1.24e3i·23-s + (2.92e3 + 1.10e3i)25-s − 4.06e3i·27-s − 270·29-s − 2.04e3·31-s + ⋯ |
L(s) = 1 | − 0.898i·3-s + (−0.983 − 0.178i)5-s − 1.21i·7-s + 0.193·9-s + 0.368·11-s − 1.12i·13-s + (−0.160 + 0.883i)15-s − 1.71i·17-s + 1.41·19-s − 1.09·21-s − 0.491i·23-s + (0.936 + 0.352i)25-s − 1.07i·27-s − 0.0596·29-s − 0.382·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.547947274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.547947274\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (55 + 10i)T \) |
good | 3 | \( 1 + 14iT - 243T^{2} \) |
| 7 | \( 1 + 158iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 148T + 1.61e5T^{2} \) |
| 13 | \( 1 + 684iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.04e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.24e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 270T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.37e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.29e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.96e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.20e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 8.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.72e4iT - 8.58e9T^{2} \) |
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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
−10.40798133918283463611814069135, −9.362097390708126835512903973996, −7.976905079686492485291398375542, −7.41206681584715330809595035596, −6.85627125866067399252101557287, −5.22660877959376333887606380084, −4.09000221554575593486135417789, −2.96737626266581221367272840611, −1.06214537871068794523823957956, −0.50331916655101393402041627034,
1.64816353048125030239897637389, 3.31417070250116321552865347618, 4.11918762105717064503094414208, 5.18043803815000671230890681206, 6.39042117936015941783695381724, 7.56174395117571142419891896144, 8.684629844601267694369339341083, 9.341510265879589143710613731559, 10.34757264904187364532269775074, 11.40152723005750657822842957554