L(s) = 1 | + (1.78 − 5.36i)2-s + (−25.6 − 19.1i)4-s + (−12.8 + 7.42i)5-s + (−15.2 − 8.83i)7-s + (−148. + 103. i)8-s + (16.9 + 82.2i)10-s + (−143. + 248. i)11-s + (42.2 + 73.1i)13-s + (−74.6 + 66.3i)14-s + (291. + 981. i)16-s + 1.60e3i·17-s − 1.66e3i·19-s + (472. + 55.6i)20-s + (1.07e3 + 1.21e3i)22-s + (2.32e3 + 4.02e3i)23-s + ⋯ |
L(s) = 1 | + (0.315 − 0.949i)2-s + (−0.801 − 0.598i)4-s + (−0.230 + 0.132i)5-s + (−0.118 − 0.0681i)7-s + (−0.820 + 0.572i)8-s + (0.0535 + 0.260i)10-s + (−0.356 + 0.618i)11-s + (0.0693 + 0.120i)13-s + (−0.101 + 0.0905i)14-s + (0.284 + 0.958i)16-s + 1.35i·17-s − 1.06i·19-s + (0.263 + 0.0311i)20-s + (0.474 + 0.533i)22-s + (0.916 + 1.58i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.850665 + 0.402831i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.850665 + 0.402831i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.78 + 5.36i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (12.8 - 7.42i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (15.2 + 8.83i)T + (8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (143. - 248. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-42.2 - 73.1i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.60e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.66e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.32e3 - 4.02e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.41e3 + 1.39e3i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-2.78e3 + 1.60e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 4.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.81e3 - 5.66e3i)T + (5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.23e4 - 7.15e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.69e3 + 2.93e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 3.10e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (1.09e4 + 1.88e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.78e4 - 3.09e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.80e4 - 1.04e4i)T + (6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 4.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (7.60e4 + 4.39e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.22e4 + 2.11e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.19e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.95e4 + 6.85e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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
−12.97081632619427836372287891537, −11.77382832000712550736110644780, −10.94580732283713102761891893589, −9.893086067762525251788928071025, −8.874055055128859558083373420644, −7.40751324344975791934574802097, −5.78194287574607761162592039895, −4.45380611600197452937841534335, −3.16653045399822261991999344046, −1.57907791984875555763381998413,
0.32807053882336509659253472431, 3.08029739306444890601566988375, 4.57336439660391216009951604978, 5.75667701369187035233069641350, 6.96674275887329566446759605182, 8.117150612154834750640916389080, 9.025190397349879330792401846854, 10.37880433785091906983668681115, 11.86679477048367681912259109759, 12.77213324113386429638692807353