| 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
−15.52314343968041215886710187322, −14.56520940099580017509565146501, −12.87246623034947686088743895832, −11.91130761546460204632821135714, −11.47910971606163952155102642835, −9.125704616579816536461780684323, −7.32345030565468155991835038390, −5.47579834116459124969990319605, −3.84643567060097433571843150918, −1.13586621627236100520069221193,
3.64124437612353444007345871687, 4.81125269373498512964341485557, 7.00214812438890266848349908494, 7.983762009783164362111964756124, 10.67229411491704566302210808152, 11.42325157483841965293710568919, 12.89246339073494809753060311571, 14.60918905746780055675025486603, 15.10869344991163992642141606196, 16.38983757362883077741397255025
