| L(s) = 1 | + (−1.65 − 1.11i)2-s + (−1.09 + 1.33i)3-s + (1.51 + 3.70i)4-s + (2.22 + 7.32i)5-s + (3.31 − 0.996i)6-s + (−0.932 + 4.68i)7-s + (1.62 − 7.83i)8-s + (−0.585 − 2.94i)9-s + (4.48 − 14.6i)10-s + (9.72 + 0.957i)11-s + (−6.61 − 2.04i)12-s + (22.7 + 6.89i)13-s + (6.78 − 6.74i)14-s + (−12.2 − 5.07i)15-s + (−11.4 + 11.1i)16-s + (5.31 + 12.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.829 − 0.557i)2-s + (−0.366 + 0.446i)3-s + (0.377 + 0.925i)4-s + (0.444 + 1.46i)5-s + (0.552 − 0.166i)6-s + (−0.133 + 0.669i)7-s + (0.202 − 0.979i)8-s + (−0.0650 − 0.326i)9-s + (0.448 − 1.46i)10-s + (0.883 + 0.0870i)11-s + (−0.551 − 0.170i)12-s + (1.74 + 0.530i)13-s + (0.484 − 0.481i)14-s + (−0.816 − 0.338i)15-s + (−0.714 + 0.699i)16-s + (0.312 + 0.754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.753512 + 0.857234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.753512 + 0.857234i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.65 + 1.11i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-2.22 - 7.32i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.932 - 4.68i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-9.72 - 0.957i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-22.7 - 6.89i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-5.31 - 12.8i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-18.9 + 10.1i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (19.1 + 12.7i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (39.8 - 3.92i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (10.1 - 10.1i)T - 961iT^{2} \) |
| 37 | \( 1 + (-4.88 - 2.61i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-42.1 - 28.1i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-36.4 - 44.4i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (9.91 + 23.9i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (68.8 + 6.77i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (19.7 + 65.1i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (9.27 - 11.3i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (38.4 + 31.5i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (70.9 + 14.1i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-87.2 + 17.3i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-33.9 + 81.8i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (12.8 + 24.0i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (67.7 + 101. i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-10.4 - 10.4i)T + 9.40e3iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14803873547331437784811617563, −10.60966150945197734896051271263, −9.543094552141096558668851970052, −9.020431116557598861307105902287, −7.72967193124058741288318846940, −6.47310163448351592389924471330, −6.06240952249710777027080731171, −3.93865912958755916230469113638, −3.08901128960173029894144134723, −1.65424773086827957807927746173,
0.799707500672031271165788600278, 1.49184129371326286344129711816, 3.98730213967660222603224267210, 5.56070981158978098938395915959, 5.91020653099616068826765253247, 7.27771087832161986740035378475, 8.077159691632920142399865339864, 9.074592396242824536098594680229, 9.622886460455438464375605879072, 10.81750312705978336399181740612
