| L(s) = 1 | + (−0.363 − 1.96i)2-s + (−1.09 + 1.33i)3-s + (−3.73 + 1.42i)4-s + (2.57 + 8.50i)5-s + (3.03 + 1.67i)6-s + (−0.889 + 4.46i)7-s + (4.16 + 6.82i)8-s + (−0.585 − 2.94i)9-s + (15.7 − 8.16i)10-s + (6.31 + 0.622i)11-s + (2.19 − 6.57i)12-s + (−16.3 − 4.95i)13-s + (9.11 + 0.123i)14-s + (−14.2 − 5.88i)15-s + (11.9 − 10.6i)16-s + (−5.53 − 13.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.181 − 0.983i)2-s + (−0.366 + 0.446i)3-s + (−0.933 + 0.357i)4-s + (0.515 + 1.70i)5-s + (0.505 + 0.279i)6-s + (−0.127 + 0.638i)7-s + (0.521 + 0.853i)8-s + (−0.0650 − 0.326i)9-s + (1.57 − 0.816i)10-s + (0.574 + 0.0565i)11-s + (0.182 − 0.547i)12-s + (−1.25 − 0.381i)13-s + (0.650 + 0.00885i)14-s + (−0.947 − 0.392i)15-s + (0.744 − 0.667i)16-s + (−0.325 − 0.785i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.218341 + 0.568524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218341 + 0.568524i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.363 + 1.96i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-2.57 - 8.50i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.889 - 4.46i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-6.31 - 0.622i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (16.3 + 4.95i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (5.53 + 13.3i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (24.4 - 13.0i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-8.87 - 5.93i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (28.6 - 2.82i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-41.1 + 41.1i)T - 961iT^{2} \) |
| 37 | \( 1 + (-52.0 - 27.8i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (59.4 + 39.6i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-5.44 - 6.63i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-2.74 - 6.63i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (86.3 + 8.50i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-10.0 - 33.1i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (27.0 - 33.0i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (29.0 + 23.8i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-15.0 - 2.99i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (69.6 - 13.8i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (34.2 - 82.6i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-65.5 - 122. i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (43.0 + 64.5i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-62.0 - 62.0i)T + 9.40e3iT^{2} \) |
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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
−11.39244506880748388692121781924, −10.52835339836235102876998849072, −9.862500773952519092759545992706, −9.253861495847907593445071577093, −7.81155706248577085798841326698, −6.64580138140190369111622611422, −5.63758645085245464136427121727, −4.31113666507007285588623097859, −3.00696508576705548759190153420, −2.23772991374427476510514207307,
0.29790755038840916753188702339, 1.60481335646962241967543088691, 4.39418937946630900589207287079, 4.85908535924942738922194781311, 6.09231255750582753992597856214, 6.85788623591673126066314843905, 8.006474516546604418687060780755, 8.831981446984716590492208081806, 9.534396191158615263562732044159, 10.52800277363843328504832638632
