| L(s) = 1 | + (0.201 + 1.39i)2-s + (0.943 − 2.06i)3-s + (−1.91 + 0.563i)4-s + (6.91 + 5.99i)5-s + (3.08 + 0.904i)6-s + (−4.33 − 6.73i)7-s + (−1.17 − 2.57i)8-s + (2.51 + 2.90i)9-s + (−7.00 + 10.8i)10-s + (−11.6 − 1.67i)11-s + (−0.646 + 4.49i)12-s + (−13.7 − 8.85i)13-s + (8.56 − 7.41i)14-s + (18.9 − 8.63i)15-s + (3.36 − 2.16i)16-s + (5.30 − 18.0i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.314 − 0.688i)3-s + (−0.479 + 0.140i)4-s + (1.38 + 1.19i)5-s + (0.513 + 0.150i)6-s + (−0.618 − 0.962i)7-s + (−0.146 − 0.321i)8-s + (0.279 + 0.322i)9-s + (−0.700 + 1.08i)10-s + (−1.05 − 0.151i)11-s + (−0.0538 + 0.374i)12-s + (−1.06 − 0.681i)13-s + (0.611 − 0.529i)14-s + (1.26 − 0.575i)15-s + (0.210 − 0.135i)16-s + (0.311 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.25266 + 0.384837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25266 + 0.384837i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.201 - 1.39i)T \) |
| 23 | \( 1 + (17.2 - 15.2i)T \) |
| good | 3 | \( 1 + (-0.943 + 2.06i)T + (-5.89 - 6.80i)T^{2} \) |
| 5 | \( 1 + (-6.91 - 5.99i)T + (3.55 + 24.7i)T^{2} \) |
| 7 | \( 1 + (4.33 + 6.73i)T + (-20.3 + 44.5i)T^{2} \) |
| 11 | \( 1 + (11.6 + 1.67i)T + (116. + 34.0i)T^{2} \) |
| 13 | \( 1 + (13.7 + 8.85i)T + (70.2 + 153. i)T^{2} \) |
| 17 | \( 1 + (-5.30 + 18.0i)T + (-243. - 156. i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 6.26i)T + (-303. + 195. i)T^{2} \) |
| 29 | \( 1 + (-4.07 - 1.19i)T + (707. + 454. i)T^{2} \) |
| 31 | \( 1 + (-5.94 - 13.0i)T + (-629. + 726. i)T^{2} \) |
| 37 | \( 1 + (11.7 - 10.1i)T + (194. - 1.35e3i)T^{2} \) |
| 41 | \( 1 + (-18.9 + 21.8i)T + (-239. - 1.66e3i)T^{2} \) |
| 43 | \( 1 + (-5.57 - 2.54i)T + (1.21e3 + 1.39e3i)T^{2} \) |
| 47 | \( 1 - 51.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-2.09 - 3.26i)T + (-1.16e3 + 2.55e3i)T^{2} \) |
| 59 | \( 1 + (-28.4 - 18.2i)T + (1.44e3 + 3.16e3i)T^{2} \) |
| 61 | \( 1 + (56.1 - 25.6i)T + (2.43e3 - 2.81e3i)T^{2} \) |
| 67 | \( 1 + (-18.1 + 2.61i)T + (4.30e3 - 1.26e3i)T^{2} \) |
| 71 | \( 1 + (2.77 + 19.2i)T + (-4.83e3 + 1.42e3i)T^{2} \) |
| 73 | \( 1 + (-79.8 + 23.4i)T + (4.48e3 - 2.88e3i)T^{2} \) |
| 79 | \( 1 + (52.2 - 81.3i)T + (-2.59e3 - 5.67e3i)T^{2} \) |
| 83 | \( 1 + (99.9 - 86.6i)T + (980. - 6.81e3i)T^{2} \) |
| 89 | \( 1 + (104. + 47.7i)T + (5.18e3 + 5.98e3i)T^{2} \) |
| 97 | \( 1 + (134. + 116. i)T + (1.33e3 + 9.31e3i)T^{2} \) |
| 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
−15.57977276991475371046601065558, −14.06302476786164623225122450980, −13.76838034683790292787631675359, −12.75209944608770817724123788017, −10.42451078275517770349435582489, −9.840997755403650575632735164866, −7.60416186522740751123257115438, −6.97159397303245723274796398334, −5.50630530689691550197235488610, −2.75395888937311358264320933953,
2.34595079785538223897675413671, 4.63258404785424889287284145641, 5.88189886192789950490782133307, 8.685792287149690185899667265128, 9.586261722653580245129170977870, 10.17882949792781066867462822623, 12.38078000220884624583482440015, 12.79939358087325906468775491565, 14.14097281966278044354073023805, 15.43041826105964739356588758061
