| L(s) = 1 | + (1.32 − 0.506i)2-s + (−0.675 + 1.59i)3-s + (1.48 − 1.33i)4-s + (−1.38 + 0.503i)5-s + (−0.0837 + 2.44i)6-s + (3.92 − 2.26i)7-s + (1.28 − 2.52i)8-s + (−2.08 − 2.15i)9-s + (−1.57 + 1.36i)10-s + (0.674 + 0.389i)11-s + (1.13 + 3.27i)12-s + (1.93 + 2.30i)13-s + (4.03 − 4.97i)14-s + (0.132 − 2.54i)15-s + (0.416 − 3.97i)16-s + (6.89 + 1.21i)17-s + ⋯ |
| L(s) = 1 | + (0.933 − 0.358i)2-s + (−0.390 + 0.920i)3-s + (0.742 − 0.669i)4-s + (−0.618 + 0.225i)5-s + (−0.0342 + 0.999i)6-s + (1.48 − 0.855i)7-s + (0.453 − 0.891i)8-s + (−0.695 − 0.718i)9-s + (−0.496 + 0.431i)10-s + (0.203 + 0.117i)11-s + (0.326 + 0.945i)12-s + (0.536 + 0.639i)13-s + (1.07 − 1.33i)14-s + (0.0340 − 0.657i)15-s + (0.104 − 0.994i)16-s + (1.67 + 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28593 - 0.0671614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28593 - 0.0671614i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 + 0.506i)T \) |
| 3 | \( 1 + (0.675 - 1.59i)T \) |
| 19 | \( 1 + (1.66 - 4.02i)T \) |
| good | 5 | \( 1 + (1.38 - 0.503i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.92 + 2.26i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.674 - 0.389i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.93 - 2.30i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-6.89 - 1.21i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.25 + 0.820i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.354 + 2.01i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.63 - 3.25i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.12iT - 37T^{2} \) |
| 41 | \( 1 + (-1.68 + 2.01i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.343 - 0.124i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.490 + 2.78i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (6.24 + 2.27i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (14.3 + 2.53i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.28 + 6.27i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.30 - 7.40i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (5.55 - 2.02i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.02 - 0.860i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (10.0 - 11.9i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.94 + 1.70i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.1 - 12.1i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.436 + 2.47i)T + (-91.1 - 33.1i)T^{2} \) |
| 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
−11.08400061138869063920071263248, −10.58477997342629284176677775869, −9.664873970068489742760571621322, −8.191989523930152724271188418387, −7.33327132078770786304783081425, −6.00376983916739856991940264424, −5.10281897788925030781714332520, −4.03630202911329500173482714718, −3.66474401232760410684039896066, −1.53423877663801019475621292572,
1.60938254671604743258993372069, 3.02701561041341335578446219806, 4.57860964974310346047534728158, 5.45237508810605532612335267253, 6.14061239880001725684149647881, 7.63688392342504730344081308323, 7.86489467918677393860175894577, 8.785996816382497326734025285878, 10.76211089543509027271531663587, 11.50771238144186185287359859652
