L(s) = 1 | + (−0.767 + 1.18i)2-s + (−0.822 − 1.82i)4-s + (−2.37 − 2.37i)5-s − 3.64i·7-s + (2.79 + 0.420i)8-s + (4.64 − 0.999i)10-s + (−0.841 − 0.841i)11-s + (−2.64 + 2.64i)13-s + (4.33 + 2.79i)14-s + (−2.64 + 2.99i)16-s + 3.06·17-s + (1.64 − 1.64i)19-s + (−2.37 + 6.28i)20-s + (1.64 − 0.354i)22-s − 7.82i·23-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.840i)2-s + (−0.411 − 0.911i)4-s + (−1.06 − 1.06i)5-s − 1.37i·7-s + (0.988 + 0.148i)8-s + (1.46 − 0.316i)10-s + (−0.253 − 0.253i)11-s + (−0.733 + 0.733i)13-s + (1.15 + 0.747i)14-s + (−0.661 + 0.749i)16-s + 0.744·17-s + (0.377 − 0.377i)19-s + (−0.531 + 1.40i)20-s + (0.350 − 0.0755i)22-s − 1.63i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.489423 - 0.305289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.489423 - 0.305289i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.767 - 1.18i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.37 + 2.37i)T + 5iT^{2} \) |
| 7 | \( 1 + 3.64iT - 7T^{2} \) |
| 11 | \( 1 + (0.841 + 0.841i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.64 - 2.64i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + (-1.64 + 1.64i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.82iT - 23T^{2} \) |
| 29 | \( 1 + (0.692 - 0.692i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.354T + 31T^{2} \) |
| 37 | \( 1 + (-4.64 - 4.64i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.43iT - 41T^{2} \) |
| 43 | \( 1 + (5.64 + 5.64i)T + 43iT^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + (5.44 + 5.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.82 - 7.82i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.64 + 4.64i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 7.29iT - 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 + (-0.841 + 0.841i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.50iT - 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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
−13.07666969320014885325588445676, −11.89519017144440000201295813339, −10.70119640640942019232898899398, −9.666616935468842334274263721944, −8.477918369327214844035293682933, −7.69612803461793922242960353307, −6.78732685017159871378205975408, −4.99308575855915356715842320538, −4.15376254940790847285444778108, −0.71150562094997107158395850868,
2.56280654812003011364367156303, 3.60227794843516931200467080590, 5.44902814528344634743732946284, 7.37005059305740434651828709466, 7.986131268028729509577513602948, 9.330828081207855854985117384678, 10.29010152048716074543334840084, 11.40936132975467720203871848397, 11.98571252310222009920302778986, 12.81519280780010891825553237631