| L(s) = 1 | + 1.73i·3-s − 5.91i·5-s + (−6.24 + 3.16i)7-s − 2.99·9-s + 1.75·11-s + 18.7i·13-s + 10.2·15-s + 23.4i·17-s + 23.0i·19-s + (−5.48 − 10.8i)21-s − 18.7·23-s − 9.97·25-s − 5.19i·27-s + 30·29-s − 8.60i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.18i·5-s + (−0.891 + 0.452i)7-s − 0.333·9-s + 0.159·11-s + 1.44i·13-s + 0.682·15-s + 1.38i·17-s + 1.21i·19-s + (−0.261 − 0.514i)21-s − 0.814·23-s − 0.398·25-s − 0.192i·27-s + 1.03·29-s − 0.277i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.495820 + 0.807505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.495820 + 0.807505i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (6.24 - 3.16i)T \) |
| good | 5 | \( 1 + 5.91iT - 25T^{2} \) |
| 11 | \( 1 - 1.75T + 121T^{2} \) |
| 13 | \( 1 - 18.7iT - 169T^{2} \) |
| 17 | \( 1 - 23.4iT - 289T^{2} \) |
| 19 | \( 1 - 23.0iT - 361T^{2} \) |
| 23 | \( 1 + 18.7T + 529T^{2} \) |
| 29 | \( 1 - 30T + 841T^{2} \) |
| 31 | \( 1 + 8.60iT - 961T^{2} \) |
| 37 | \( 1 + 70.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 97.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 69.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.43iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 51.7iT - 9.40e3T^{2} \) |
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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
−12.00814593433989774902594468184, −10.49458354920954172445125942216, −9.707448190180284221990921236981, −8.866785256678216873148110770948, −8.257159607369387551122001164480, −6.58402543411738906285473566537, −5.73250939295212492867669574900, −4.51098842658064343494365140586, −3.62613582084290743852940704694, −1.74625242334039098485123549288,
0.43140287921707291468523572935, 2.66223153185531742375198214655, 3.38507408596804306107339450768, 5.19070318853289860044026913612, 6.52956749643250674889923863185, 6.99395024616212978646593103072, 7.952595738912013079993976530926, 9.276677432099027766956975311156, 10.28891089082049676467549625448, 10.89856637676266555178073341202
