| L(s) = 1 | + (−6.92 − 4i)2-s + (16.1 − 28.0i)3-s + (31.9 + 55.4i)4-s − 211. i·5-s + (−224. + 129. i)6-s + (635. − 367. i)7-s − 511. i·8-s + (570. + 987. i)9-s + (−845. + 1.46e3i)10-s + (−7.44e3 − 4.29e3i)11-s + 2.07e3·12-s + (−6.92e3 − 3.84e3i)13-s − 5.87e3·14-s + (−5.92e3 − 3.41e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−8.16e3 − 1.41e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.345 − 0.599i)3-s + (0.249 + 0.433i)4-s − 0.756i·5-s + (−0.423 + 0.244i)6-s + (0.700 − 0.404i)7-s − 0.353i·8-s + (0.260 + 0.451i)9-s + (−0.267 + 0.463i)10-s + (−1.68 − 0.973i)11-s + 0.345·12-s + (−0.874 − 0.484i)13-s − 0.572·14-s + (−0.453 − 0.261i)15-s + (−0.125 + 0.216i)16-s + (−0.403 − 0.698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.296839 - 1.04437i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.296839 - 1.04437i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.92 + 4i)T \) |
| 13 | \( 1 + (6.92e3 + 3.84e3i)T \) |
| good | 3 | \( 1 + (-16.1 + 28.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + 211. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (-635. + 367. i)T + (4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (7.44e3 + 4.29e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (8.16e3 + 1.41e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.03e3 + 1.75e3i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.26e4 - 5.65e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-5.62e4 + 9.75e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.76e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-4.32e5 - 2.49e5i)T + (4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (2.50e5 + 1.44e5i)T + (9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.01e5 + 3.48e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + 6.49e4iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 2.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.22e6 + 1.28e6i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.69e5 - 4.66e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-7.71e5 - 4.45e5i)T + (3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-3.42e6 + 1.97e6i)T + (4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 2.77e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 4.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.61e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (6.82e6 + 3.93e6i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-7.91e6 + 4.57e6i)T + (4.03e13 - 6.99e13i)T^{2} \) |
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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
−15.63469880382115108456306002176, −13.69103267235617757928906875427, −12.93056764941672012763999790364, −11.38411312835472227564525086027, −10.05269645704284902528632322311, −8.264986582808183660861897489083, −7.61808911757209880837922086763, −5.04046010658281091793793212243, −2.43473850845397580425618324778, −0.63448499344094674100655244111,
2.41730333207208235313150591332, 4.83118570810544310859604166697, 6.90995703757590682443021294737, 8.308360787232821894525869692568, 9.846035557171850315241763091188, 10.72839380013227193584852136119, 12.49195245710228716771529261491, 14.59848973347422541882932115332, 15.05109209776244580925812121470, 16.20358752160794225044166148676
