L(s) = 1 | − 26.2i·3-s + (48.8 − 27.1i)5-s + 23.9i·7-s − 447.·9-s − 704.·11-s + 1.03e3i·13-s + (−713. − 1.28e3i)15-s − 217. i·17-s − 1.29e3·19-s + 628.·21-s − 2.80e3i·23-s + (1.65e3 − 2.65e3i)25-s + 5.36e3i·27-s − 4.90e3·29-s − 415.·31-s + ⋯ |
L(s) = 1 | − 1.68i·3-s + (0.874 − 0.485i)5-s + 0.184i·7-s − 1.84·9-s − 1.75·11-s + 1.70i·13-s + (−0.818 − 1.47i)15-s − 0.182i·17-s − 0.819·19-s + 0.311·21-s − 1.10i·23-s + (0.528 − 0.849i)25-s + 1.41i·27-s − 1.08·29-s − 0.0776·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4524778981\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4524778981\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-48.8 + 27.1i)T \) |
good | 3 | \( 1 + 26.2iT - 243T^{2} \) |
| 7 | \( 1 - 23.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 704.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 217. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.29e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.80e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 415.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.45e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.64e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.37e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.23e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.03e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.79e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.27e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.25e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.41e4iT - 8.58e9T^{2} \) |
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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
−11.62200259239647762724669095347, −10.34774117735005979061035383439, −8.963349509549293381657959485938, −8.126837888802368396124647727760, −6.94944339939491738916020202562, −6.11571835263399871312512753877, −4.92048435777121072920779766738, −2.46151803208059319307785816716, −1.74583305100143525204530096768, −0.13357109662032687206075867166,
2.57080955443150374792066600907, 3.62360032487593960610232107051, 5.25261260286411551591113747793, 5.65018164058507792122257378531, 7.59417508110628343891907202158, 8.812994984991626008192708359761, 10.04792757167686880544323252281, 10.39244493579380310380449687638, 11.06048085948890916842563428781, 12.87132038389746153759070213321