L(s) = 1 | + (−0.821 − 5.59i)2-s + (−30.6 + 9.19i)4-s − 8.15i·5-s + 63.0i·7-s + (76.6 + 164. i)8-s + (−45.6 + 6.69i)10-s + 442.·11-s − 76.3·13-s + (352. − 51.7i)14-s + (855. − 563. i)16-s + 643. i·17-s − 2.18e3i·19-s + (74.9 + 250. i)20-s + (−363. − 2.47e3i)22-s + 2.73e3·23-s + ⋯ |
L(s) = 1 | + (−0.145 − 0.989i)2-s + (−0.957 + 0.287i)4-s − 0.145i·5-s + 0.485i·7-s + (0.423 + 0.906i)8-s + (−0.144 + 0.0211i)10-s + 1.10·11-s − 0.125·13-s + (0.480 − 0.0705i)14-s + (0.835 − 0.550i)16-s + 0.540i·17-s − 1.39i·19-s + (0.0419 + 0.139i)20-s + (−0.160 − 1.09i)22-s + 1.07·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.29213 - 0.961521i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29213 - 0.961521i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.821 + 5.59i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 8.15iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 63.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 442.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 76.3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 643. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.50e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.50e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 4.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.09e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 8.35e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.28e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.10e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.73e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.32e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 6.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.20e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e5T + 8.58e9T^{2} \) |
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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
−12.40716168213724077019879671066, −11.51331462490891912138989972984, −10.58017486562462774037398654384, −9.200780054091958455090881057908, −8.722312827371494492886571217467, −7.01538261720994299705501399016, −5.30903094319330509447774640734, −3.98801802876030196623544225320, −2.49821576032129306700121063929, −0.904548895537723653401249760891,
1.05500315223204899615386919366, 3.65640238071442402869230268226, 4.99176224836901845403562433160, 6.41970998554280741308526315743, 7.27004861749847874748330227031, 8.526730000716856583507385684688, 9.552607492608964470162051639182, 10.62413480777224323126866388338, 12.06494512806177720463634117841, 13.23624856663854824511337568441