| L(s) = 1 | + (−0.104 + 0.994i)2-s + (−0.845 − 1.51i)3-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (1.59 − 0.682i)6-s + (3.24 − 3.60i)7-s + (0.309 − 0.951i)8-s + (−1.57 + 2.55i)9-s + 0.999·10-s + (2.72 + 1.88i)11-s + (0.512 + 1.65i)12-s + (−0.947 + 0.421i)13-s + (3.24 + 3.60i)14-s + (−1.41 + 0.998i)15-s + (0.913 + 0.406i)16-s + (5.36 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (−0.487 − 0.872i)3-s + (−0.489 − 0.103i)4-s + (−0.0467 − 0.444i)5-s + (0.649 − 0.278i)6-s + (1.22 − 1.36i)7-s + (0.109 − 0.336i)8-s + (−0.523 + 0.851i)9-s + 0.316·10-s + (0.822 + 0.569i)11-s + (0.147 + 0.477i)12-s + (−0.262 + 0.116i)13-s + (0.868 + 0.964i)14-s + (−0.365 + 0.257i)15-s + (0.228 + 0.101i)16-s + (1.30 − 0.944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14348 - 0.755969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14348 - 0.755969i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.845 + 1.51i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 11 | \( 1 + (-2.72 - 1.88i)T \) |
| good | 7 | \( 1 + (-3.24 + 3.60i)T + (-0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (0.947 - 0.421i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-5.36 + 3.89i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.19 - 3.67i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.73 + 3.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.51 + 5.01i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (6.89 - 3.07i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.14 - 3.51i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (7.47 + 8.30i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (4.30 + 7.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.19 - 0.891i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.579 - 0.420i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-11.9 - 2.54i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.04 - 1.35i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.20 + 3.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.537 - 0.390i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.15 - 6.64i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.305 + 2.91i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (6.86 + 3.05i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 - 9.51T + 89T^{2} \) |
| 97 | \( 1 + (0.721 - 6.86i)T + (-94.8 - 20.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.919283397161751766972202043667, −8.612351116565314511946909849324, −7.975115053794354389649908575512, −7.21626747375033917774742812118, −6.77247303683752675201204345079, −5.43913210372614377319846657161, −4.83666811916602207716142428906, −3.85126596052184670988480406932, −1.75172381767402040550799729097, −0.790835312997144258272896470837,
1.47682414043037520923953447564, 2.88820074508264609282865301523, 3.77546717013070731665386691835, 4.97481074665937227618450068224, 5.51228913262658684167160483254, 6.50454597209647405271429093931, 8.011647561627535318740086272937, 8.698057895618628188402976935492, 9.402868883109090262267584504132, 10.23043820124484070866803578903
