| L(s) = 1 | + (−0.939 − 0.342i)3-s + (−0.189 − 1.07i)5-s + (1.46 + 0.843i)7-s + (0.766 + 0.642i)9-s + (0.301 − 0.174i)11-s + (−0.580 − 1.59i)13-s + (−0.189 + 1.07i)15-s + (3.79 − 3.18i)17-s + (−3.18 + 2.97i)19-s + (−1.08 − 1.29i)21-s + (1.82 + 0.321i)23-s + (3.57 − 1.30i)25-s + (−0.500 − 0.866i)27-s + (2.36 − 2.81i)29-s + (−1.63 + 2.83i)31-s + ⋯ |
| L(s) = 1 | + (−0.542 − 0.197i)3-s + (−0.0847 − 0.480i)5-s + (0.552 + 0.318i)7-s + (0.255 + 0.214i)9-s + (0.0909 − 0.0525i)11-s + (−0.161 − 0.442i)13-s + (−0.0489 + 0.277i)15-s + (0.920 − 0.772i)17-s + (−0.730 + 0.682i)19-s + (−0.236 − 0.282i)21-s + (0.379 + 0.0669i)23-s + (0.715 − 0.260i)25-s + (−0.0962 − 0.166i)27-s + (0.438 − 0.522i)29-s + (−0.293 + 0.508i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16630 - 0.632881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16630 - 0.632881i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (3.18 - 2.97i)T \) |
| good | 5 | \( 1 + (0.189 + 1.07i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 0.843i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.301 + 0.174i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.580 + 1.59i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.79 + 3.18i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.82 - 0.321i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.36 + 2.81i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.42iT - 37T^{2} \) |
| 41 | \( 1 + (-1.92 + 5.27i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.230 + 0.0405i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.79 + 8.09i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (2.24 + 0.396i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.47 + 5.43i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.18 + 6.74i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.56 + 2.99i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.08 + 6.13i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 4.03i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.99 + 0.725i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.17 - 1.25i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.87 - 5.13i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.92 - 4.67i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.08103478824777759595338130643, −9.047468910229210287899222216182, −8.254003104177620811303358646889, −7.47760451764071675949993244766, −6.47547084214592560671208674758, −5.42118408101369618772535695929, −4.91634688450533976283110519570, −3.66098700729569654546298870678, −2.20524702523471925144315608987, −0.793601406407526919532228023873,
1.27511605130550077615382708298, 2.82394682574092882985304742112, 4.08415009203712125953432215348, 4.86328254904666988179947716389, 5.94280395638366788003303920875, 6.79944669491924637079990940587, 7.57345804871470879882302552300, 8.558071331372337157598794740056, 9.486525034954028764674223862746, 10.46248027081482968801508395552
