L(s) = 1 | + 25.4i·2-s + 123. i·3-s − 389.·4-s + 710. i·5-s − 3.15e3·6-s − 4.24e3i·7-s − 3.40e3i·8-s − 8.81e3·9-s − 1.80e4·10-s + 1.92e4·11-s − 4.83e4i·12-s − 2.45e4·13-s + 1.07e5·14-s − 8.81e4·15-s − 1.32e4·16-s − 1.42e5·17-s + ⋯ |
L(s) = 1 | + 1.58i·2-s + 1.53i·3-s − 1.52·4-s + 1.13i·5-s − 2.43·6-s − 1.76i·7-s − 0.831i·8-s − 1.34·9-s − 1.80·10-s + 1.31·11-s − 2.33i·12-s − 0.859·13-s + 2.80·14-s − 1.74·15-s − 0.202·16-s − 1.70·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.871680 - 0.399615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.871680 - 0.399615i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-2.23e6 - 2.59e6i)T \) |
good | 2 | \( 1 - 25.4iT - 256T^{2} \) |
| 3 | \( 1 - 123. iT - 6.56e3T^{2} \) |
| 5 | \( 1 - 710. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 4.24e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.92e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.45e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.42e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.52e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.69e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 3.50e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.67e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 9.65e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 4.61e6T + 7.98e12T^{2} \) |
| 47 | \( 1 - 2.55e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.21e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 3.24e6T + 1.46e14T^{2} \) |
| 61 | \( 1 - 1.17e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 3.62e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.36e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.61e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 3.75e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.67e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 4.02e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 3.96e6T + 7.83e15T^{2} \) |
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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
−15.05783279023048258823153003029, −14.58873003307903904390750146224, −13.76623886725786842991863895758, −11.10971220339878630207230832984, −10.23814434707237441217146628907, −9.056882349721502554505643543450, −7.31446425940763745825289546210, −6.51520407546007714473659563567, −4.64599661469411787708767311592, −3.75441991550348660260973683199,
0.36134767044291127805143051293, 1.67974681237816226234701169223, 2.51426928698628307377852516584, 4.83342825553265379481988733387, 6.62887914516877475529292476740, 8.799387349138996081127632419030, 9.124050423738358646040167337305, 11.51777829307013139244506996303, 12.05551962235675017627588184640, 12.75361499413393733191964420858