| L(s) = 1 | + (−1.61 + 1.17i)2-s + (−3.04 − 9.37i)3-s + (1.23 − 3.80i)4-s + (1.30 + 0.951i)5-s + (15.9 + 11.5i)6-s + (4.57 − 14.0i)7-s + (2.47 + 7.60i)8-s + (−56.7 + 41.2i)9-s − 3.23·10-s + (35.5 + 7.99i)11-s − 39.4·12-s + (41.1 − 29.9i)13-s + (9.14 + 28.1i)14-s + (4.92 − 15.1i)15-s + (−12.9 − 9.40i)16-s + (48.7 + 35.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.586 − 1.80i)3-s + (0.154 − 0.475i)4-s + (0.117 + 0.0850i)5-s + (1.08 + 0.788i)6-s + (0.246 − 0.759i)7-s + (0.109 + 0.336i)8-s + (−2.10 + 1.52i)9-s − 0.102·10-s + (0.975 + 0.219i)11-s − 0.948·12-s + (0.878 − 0.638i)13-s + (0.174 + 0.537i)14-s + (0.0848 − 0.261i)15-s + (−0.202 − 0.146i)16-s + (0.696 + 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.543695 - 0.507145i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.543695 - 0.507145i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 - 1.17i)T \) |
| 11 | \( 1 + (-35.5 - 7.99i)T \) |
| good | 3 | \( 1 + (3.04 + 9.37i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-1.30 - 0.951i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-4.57 + 14.0i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-41.1 + 29.9i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-48.7 - 35.4i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (8.16 + 25.1i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 29.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (46.4 - 142. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (23.7 - 17.2i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (69.4 - 213. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-36.9 - 113. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (54.0 + 166. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-51.1 + 37.1i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (80.9 - 249. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-573. - 416. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 151.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (853. + 620. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (9.74 - 29.9i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (52.6 - 38.2i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-186. - 135. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 618.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-889. + 646. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−17.48889833517301111251903578414, −16.56140133071725943875160461871, −14.43425228110415364384607791989, −13.36090016500174947128266615355, −11.99724676293981293732439633659, −10.67772981168577718714218575810, −8.348982683636203967668500566687, −7.15034933082561905444553428926, −6.00880382143101634036294082962, −1.21742615757564394572398109215,
3.86481701959384562163662199456, 5.77692110461709407696167239516, 8.846824525044530794240294505285, 9.656526450348949311601438725277, 11.11046832442640062795435444615, 11.88397381502875296852872429535, 14.37665718757283807328225623726, 15.68424085478847796279638277759, 16.57372336826201576297104095579, 17.54711040795928208514253046078
