| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.605 + 1.32i)3-s + (−0.959 + 0.281i)4-s + (2.76 − 3.18i)5-s + (−1.39 − 0.410i)6-s + (0.841 − 0.540i)7-s + (−0.415 − 0.909i)8-s + (0.571 + 0.660i)9-s + (3.54 + 2.28i)10-s + (0.625 − 4.34i)11-s + (0.207 − 1.44i)12-s + (5.54 + 3.56i)13-s + (0.654 + 0.755i)14-s + (2.55 + 5.59i)15-s + (0.841 − 0.540i)16-s + (−4.58 − 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.349 + 0.765i)3-s + (−0.479 + 0.140i)4-s + (1.23 − 1.42i)5-s + (−0.571 − 0.167i)6-s + (0.317 − 0.204i)7-s + (−0.146 − 0.321i)8-s + (0.190 + 0.220i)9-s + (1.12 + 0.721i)10-s + (0.188 − 1.31i)11-s + (0.0599 − 0.416i)12-s + (1.53 + 0.987i)13-s + (0.175 + 0.201i)14-s + (0.659 + 1.44i)15-s + (0.210 − 0.135i)16-s + (−1.11 − 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40480 + 0.590912i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40480 + 0.590912i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (2.09 - 4.31i)T \) |
| good | 3 | \( 1 + (0.605 - 1.32i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (-2.76 + 3.18i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.625 + 4.34i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.54 - 3.56i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (4.58 + 1.34i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (2.03 - 0.596i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.00 - 0.293i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.59 - 5.69i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.72 - 3.14i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (0.220 - 0.254i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.376 + 0.823i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 + (-7.30 + 4.69i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (11.8 + 7.59i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.31 + 5.06i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.41 - 9.83i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.197 + 1.37i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (8.65 - 2.54i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (5.35 + 3.44i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-1.91 - 2.21i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (4.84 - 10.6i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (5.34 - 6.16i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.61559230160748790172765599428, −10.74286638611406854351147880076, −9.645495525640826183235190974989, −8.842441450591985795902767272971, −8.347885434408438359136675374779, −6.49793358882212884322253716944, −5.75765467690808655664232997767, −4.84564326446758506666646191984, −4.01404185351479532152220677803, −1.47337616915995603358773740176,
1.67266761427618440052383123275, 2.60506985384366762052913044799, 4.19276346947481934782266076028, 5.95953111391755377074168090616, 6.38083044394848754078245673066, 7.47841612572031041304701561334, 8.898944986723962760786344147966, 9.994650772171180988055741994333, 10.63897918632981767392882029971, 11.36956861831205785871725322781
