L(s) = 1 | + (1.08 + 0.905i)2-s − i·3-s + (0.360 + 1.96i)4-s − 3.71i·5-s + (0.905 − 1.08i)6-s + 1.13·7-s + (−1.38 + 2.46i)8-s − 9-s + (3.36 − 4.03i)10-s + 1.53i·11-s + (1.96 − 0.360i)12-s − 4.95i·13-s + (1.23 + 1.02i)14-s − 3.71·15-s + (−3.73 + 1.41i)16-s + 7.22·17-s + ⋯ |
L(s) = 1 | + (0.768 + 0.640i)2-s − 0.577i·3-s + (0.180 + 0.983i)4-s − 1.66i·5-s + (0.369 − 0.443i)6-s + 0.429·7-s + (−0.491 + 0.871i)8-s − 0.333·9-s + (1.06 − 1.27i)10-s + 0.463i·11-s + (0.567 − 0.104i)12-s − 1.37i·13-s + (0.330 + 0.275i)14-s − 0.959·15-s + (−0.934 + 0.354i)16-s + 1.75·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15982 - 0.566768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15982 - 0.566768i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.08 - 0.905i)T \) |
| 3 | \( 1 + iT \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3.71iT - 5T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 - 1.53iT - 11T^{2} \) |
| 13 | \( 1 + 4.95iT - 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 + 4.69iT - 19T^{2} \) |
| 29 | \( 1 - 2.85iT - 29T^{2} \) |
| 31 | \( 1 - 0.770T + 31T^{2} \) |
| 37 | \( 1 + 6.52iT - 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 - 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 2.32iT - 61T^{2} \) |
| 67 | \( 1 - 6.38iT - 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 1.06T + 73T^{2} \) |
| 79 | \( 1 - 1.16T + 79T^{2} \) |
| 83 | \( 1 - 8.10iT - 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 6.44T + 97T^{2} \) |
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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
−10.98490292734818422309241443700, −9.560072755234582987200185548955, −8.585303235626451744548007909413, −7.929350620825303774028711638669, −7.26279824384983807824938620215, −5.74958158608666920016950027419, −5.27829219309983695356839404044, −4.36752887514451720161033798336, −2.91277408257629121907742397390, −1.13751427043507403846185834568,
1.94881213559745427135664906848, 3.25833368803192943742350096793, 3.82496140639418690290753679989, 5.18021287721475594052421692487, 6.15340530099657662579953245451, 6.95587727527232243833720704362, 8.173097069786320686200734499055, 9.665533551258577244243935201904, 10.13399308981957956124536722364, 10.95645621568797695468907897012