L(s) = 1 | + (0.132 − 0.0766i)3-s + (1.14 − 0.658i)5-s + 1.40i·7-s + (−1.48 + 2.57i)9-s + 3.27·11-s + (−2.31 + 4.01i)13-s + (0.100 − 0.174i)15-s + (−2.24 − 3.89i)17-s + (2.56 + 3.52i)19-s + (0.107 + 0.186i)21-s + (0.100 + 0.0582i)23-s + (−1.63 + 2.82i)25-s + 0.915i·27-s + (−3.34 + 5.78i)29-s + 2.43·31-s + ⋯ |
L(s) = 1 | + (0.0766 − 0.0442i)3-s + (0.509 − 0.294i)5-s + 0.531i·7-s + (−0.496 + 0.859i)9-s + 0.987·11-s + (−0.642 + 1.11i)13-s + (0.0260 − 0.0451i)15-s + (−0.545 − 0.945i)17-s + (0.588 + 0.808i)19-s + (0.0235 + 0.0407i)21-s + (0.0210 + 0.0121i)23-s + (−0.326 + 0.565i)25-s + 0.176i·27-s + (−0.620 + 1.07i)29-s + 0.437·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579618384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579618384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.56 - 3.52i)T \) |
good | 3 | \( 1 + (-0.132 + 0.0766i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.14 + 0.658i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 1.40iT - 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 + (2.31 - 4.01i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.24 + 3.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.100 - 0.0582i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 8.87T + 37T^{2} \) |
| 41 | \( 1 + (-5.96 + 3.44i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.01 - 6.95i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.69 + 4.44i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.47 - 4.27i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.19 + 5.30i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.55 - 5.51i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 0.520i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.84 + 4.93i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.104 - 0.181i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.20 - 14.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + (5.18 + 2.99i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.71 - 3.30i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.548810563870828860158557806808, −9.256747653360887794770517434911, −8.434449450836502374179725455676, −7.36639593914860120885971280394, −6.64228272609508852440019691935, −5.53290396673259178359484583666, −4.98279419388309752196870290848, −3.79870210011844501406165356661, −2.48264214429006509080691881866, −1.60617561577685533122359349594,
0.66816546031832207293597835724, 2.24780658688470893280665502259, 3.37519637860659201327757569069, 4.21908292452344205049510752808, 5.46335049658591023259092332433, 6.28319165005619912484162650470, 6.94669652978495273354272717692, 7.956466878193406206774016769070, 8.842403137085033582122745335130, 9.626950972907790836517383085231