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
\(L(3)\) |
\(\approx\) |
\(0.291469 + 0.216891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.291469 + 0.216891i\) |
\(L(\frac{7}{2})\) |
|
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
−12.77289003137173566962245761174, −12.08115575269090284596758989672, −10.61459075301398949434673082548, −10.23938149051712404902894630470, −8.782572172909747557521524072168, −7.71091522652835688910186813090, −5.82924293301247829492002238329, −4.55814994966356496398061287484, −3.21720840408391944515033280059, −1.60304599912431397563386432224,
0.13036777292092968409116174284, 2.79994953648595759278393695543, 4.61388191229714509083594842881, 5.66791644397259407882318292826, 6.93967237991142753612565572291, 8.004046209897939829077621753315, 9.069014215232798261700608450049, 10.19510855525732366113680085231, 11.64324948022896362440241330150, 12.88298058723591011946992996080