| L(s) = 1 | + (4.87 + 2.86i)2-s + (−5.07 + 8.78i)3-s + (15.5 + 27.9i)4-s + (−73.9 + 42.6i)5-s + (−49.9 + 28.2i)6-s + (80.3 − 101. i)7-s + (−4.39 + 180. i)8-s + (70.0 + 121. i)9-s + (−482. − 3.91i)10-s + (256. + 148. i)11-s + (−324. − 5.26i)12-s − 324. i·13-s + (683. − 265. i)14-s − 866. i·15-s + (−540. + 869. i)16-s + (1.37e3 + 795. i)17-s + ⋯ |
| L(s) = 1 | + (0.861 + 0.506i)2-s + (−0.325 + 0.563i)3-s + (0.485 + 0.874i)4-s + (−1.32 + 0.763i)5-s + (−0.566 + 0.320i)6-s + (0.620 − 0.784i)7-s + (−0.0242 + 0.999i)8-s + (0.288 + 0.498i)9-s + (−1.52 − 0.0123i)10-s + (0.640 + 0.369i)11-s + (−0.650 − 0.0105i)12-s − 0.533i·13-s + (0.932 − 0.361i)14-s − 0.994i·15-s + (−0.527 + 0.849i)16-s + (1.15 + 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.969562 + 1.61393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.969562 + 1.61393i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.87 - 2.86i)T \) |
| 7 | \( 1 + (-80.3 + 101. i)T \) |
| good | 3 | \( 1 + (5.07 - 8.78i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (73.9 - 42.6i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-256. - 148. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 324. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-1.37e3 - 795. i)T + (7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (587. + 1.01e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-4.09e3 + 2.36e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 4.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (987. - 1.71e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-669. - 1.15e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 9.20e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.11e3 - 1.92e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-4.12e3 + 7.14e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.06e4 + 3.58e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.17e4 + 6.79e3i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.42e4 - 1.40e4i)T + (6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.32e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.16e4 + 6.71e3i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (9.71e3 - 5.61e3i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-6.56e4 + 3.79e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 7.93e4iT - 8.58e9T^{2} \) |
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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
−16.38983757362883077741397255025, −15.10869344991163992642141606196, −14.60918905746780055675025486603, −12.89246339073494809753060311571, −11.42325157483841965293710568919, −10.67229411491704566302210808152, −7.983762009783164362111964756124, −7.00214812438890266848349908494, −4.81125269373498512964341485557, −3.64124437612353444007345871687,
1.13586621627236100520069221193, 3.84643567060097433571843150918, 5.47579834116459124969990319605, 7.32345030565468155991835038390, 9.125704616579816536461780684323, 11.47910971606163952155102642835, 11.91130761546460204632821135714, 12.87246623034947686088743895832, 14.56520940099580017509565146501, 15.52314343968041215886710187322
