L(s) = 1 | + (−1.01 − 4.78i)2-s + (3.24 + 4.05i)3-s + (−14.5 + 6.45i)4-s + (−5.06 − 9.96i)5-s + (16.1 − 19.6i)6-s + (11.0 − 6.37i)7-s + (22.6 + 31.1i)8-s + (−5.93 + 26.3i)9-s + (−42.4 + 34.3i)10-s + (−32.9 + 6.99i)11-s + (−73.2 − 37.9i)12-s + (−0.362 + 1.70i)13-s + (−41.6 − 46.2i)14-s + (23.9 − 52.9i)15-s + (40.9 − 45.4i)16-s + (−75.1 − 103. i)17-s + ⋯ |
L(s) = 1 | + (−0.359 − 1.69i)2-s + (0.624 + 0.780i)3-s + (−1.81 + 0.807i)4-s + (−0.453 − 0.891i)5-s + (1.09 − 1.33i)6-s + (0.595 − 0.344i)7-s + (1.00 + 1.37i)8-s + (−0.219 + 0.975i)9-s + (−1.34 + 1.08i)10-s + (−0.902 + 0.191i)11-s + (−1.76 − 0.912i)12-s + (−0.00772 + 0.0363i)13-s + (−0.795 − 0.883i)14-s + (0.412 − 0.910i)15-s + (0.639 − 0.710i)16-s + (−1.07 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00670 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00670 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0865817 + 0.0860033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0865817 + 0.0860033i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.24 - 4.05i)T \) |
| 5 | \( 1 + (5.06 + 9.96i)T \) |
good | 2 | \( 1 + (1.01 + 4.78i)T + (-7.30 + 3.25i)T^{2} \) |
| 7 | \( 1 + (-11.0 + 6.37i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (32.9 - 6.99i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (0.362 - 1.70i)T + (-2.00e3 - 893. i)T^{2} \) |
| 17 | \( 1 + (75.1 + 103. i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (93.2 - 67.7i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-42.4 + 38.2i)T + (1.27e3 - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (19.2 - 183. i)T + (-2.38e4 - 5.07e3i)T^{2} \) |
| 31 | \( 1 + (-6.00 - 57.1i)T + (-2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (-2.72 + 0.884i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-5.01 - 1.06i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (177. - 102. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-212. - 22.2i)T + (1.01e5 + 2.15e4i)T^{2} \) |
| 53 | \( 1 + (-180. + 248. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (775. + 164. i)T + (1.87e5 + 8.35e4i)T^{2} \) |
| 61 | \( 1 + (-429. + 91.2i)T + (2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (337. - 35.4i)T + (2.94e5 - 6.25e4i)T^{2} \) |
| 71 | \( 1 + (772. + 560. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (781. + 254. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-91.6 + 872. i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (107. - 240. i)T + (-3.82e5 - 4.24e5i)T^{2} \) |
| 89 | \( 1 + (-359. + 1.10e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.33e3 - 140. i)T + (8.92e5 + 1.89e5i)T^{2} \) |
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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
−10.91274717064602489558232013121, −10.31613387219272415055840389070, −9.165371859018923198499941607723, −8.640381151419249207918739769618, −7.65026884445531509225645836369, −4.89234086419489808153118978937, −4.41263959591153951582441953266, −3.09319981025149691175777467038, −1.82338455969408966940876894671, −0.05027605016384276030105979361,
2.39326380905585489331695744017, 4.25882849548837660231178247566, 5.86525004644693108827437641130, 6.65949105637355484295752965734, 7.56527003309583218072427937454, 8.282533193408802210960008176251, 8.902286540242987153692016540017, 10.41294945631994905369284682945, 11.52750307780690726895300899377, 13.01670208062227901603193305784