L(s) = 1 | + 16.5i·3-s − 25i·5-s − 55.4·7-s − 31.1·9-s − 153. i·11-s − 57.3i·13-s + 413.·15-s + 1.78e3·17-s + 1.59e3i·19-s − 918. i·21-s − 4.11e3·23-s − 625·25-s + 3.50e3i·27-s + 5.00e3i·29-s − 1.82e3·31-s + ⋯ |
L(s) = 1 | + 1.06i·3-s − 0.447i·5-s − 0.427·7-s − 0.128·9-s − 0.382i·11-s − 0.0940i·13-s + 0.475·15-s + 1.49·17-s + 1.01i·19-s − 0.454i·21-s − 1.62·23-s − 0.200·25-s + 0.925i·27-s + 1.10i·29-s − 0.340·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9993862735\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9993862735\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 25iT \) |
good | 3 | \( 1 - 16.5iT - 243T^{2} \) |
| 7 | \( 1 + 55.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 153. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 57.3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.59e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.80e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 5.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.25e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.74e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 17.5iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.88e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.97e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.12e4T + 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03865797616050680675212079218, −10.02196082969068156534199719976, −9.686154148378528157012371475686, −8.510176055358193198153043337906, −7.58572070286525733314143608078, −6.07751607676997700517780841570, −5.24197053279454247530523240591, −4.04592064682862567714265384348, −3.26554012045303711294523483816, −1.40603603756712659564852137878,
0.26395195007646110864636698218, 1.62856870754518905643971614165, 2.77046799936795646809726219999, 4.12184225586284327058626147355, 5.69387792845239613347957017798, 6.60097563407526832286117773503, 7.42842025337638863372704071571, 8.156505864127968206193517601330, 9.604467405970492152734004193658, 10.23930529873486129525501043687