| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−1.15 + 0.664i)3-s + (−1.73 − i)4-s + (4.92 + 0.876i)5-s + (−0.486 − 1.81i)6-s + (1.39 + 5.19i)7-s + (2 − 1.99i)8-s + (−3.61 + 6.26i)9-s + (−2.99 + 6.40i)10-s + (0.498 − 1.86i)11-s + 2.65·12-s + (−6.89 + 11.0i)13-s − 7.60·14-s + (−6.24 + 2.26i)15-s + (1.99 + 3.46i)16-s + (−7.39 + 12.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.383 + 0.221i)3-s + (−0.433 − 0.250i)4-s + (0.984 + 0.175i)5-s + (−0.0810 − 0.302i)6-s + (0.198 + 0.742i)7-s + (0.250 − 0.249i)8-s + (−0.402 + 0.696i)9-s + (−0.299 + 0.640i)10-s + (0.0453 − 0.169i)11-s + 0.221·12-s + (−0.530 + 0.847i)13-s − 0.543·14-s + (−0.416 + 0.150i)15-s + (0.124 + 0.216i)16-s + (−0.434 + 0.753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.558194 + 0.994598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.558194 + 0.994598i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 5 | \( 1 + (-4.92 - 0.876i)T \) |
| 13 | \( 1 + (6.89 - 11.0i)T \) |
| good | 3 | \( 1 + (1.15 - 0.664i)T + (4.5 - 7.79i)T^{2} \) |
| 7 | \( 1 + (-1.39 - 5.19i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-0.498 + 1.86i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (7.39 - 12.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.436 - 1.62i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (4.86 + 8.43i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-18.1 - 31.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-40.0 + 40.0i)T - 961iT^{2} \) |
| 37 | \( 1 + (16.7 + 4.48i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-47.0 - 12.6i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (-2.51 + 4.35i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-10.7 + 10.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 48.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (69.1 - 18.5i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 + 64.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.1 + 82.7i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (0.0344 + 0.128i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (12.1 - 12.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 99.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-99.6 - 99.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (22.6 - 84.3i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (123. - 33.1i)T + (8.14e3 - 4.70e3i)T^{2} \) |
| 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
−13.70727323988804396896049155432, −12.47263980857069374818232232988, −11.21255744007113957456387907477, −10.19783349833897441283726014056, −9.159394263065180064821373046990, −8.184692195912293459652330517617, −6.61696828907096255093536916675, −5.74060606813063365222608259325, −4.69919818704175168608691313362, −2.24489784157711072919574645431,
0.907786783198373902483135861451, 2.79503886235755704287205725602, 4.64575229296843516173265252803, 5.95473195867786452843957305054, 7.27415091579191148057273984017, 8.744484144847032524142840137346, 9.809467690082053737586722639715, 10.57523979571878870543777101114, 11.75250134124367444639942414305, 12.60291398286677776886236742132
