L(s) = 1 | + (0.406 − 1.35i)2-s + (0.130 − 0.991i)3-s + (−1.66 − 1.10i)4-s + (−0.467 − 3.55i)5-s + (−1.28 − 0.580i)6-s + (1.00 + 2.44i)7-s + (−2.17 + 1.81i)8-s + (−0.965 − 0.258i)9-s + (−5.00 − 0.812i)10-s + (−1.65 − 2.15i)11-s + (−1.31 + 1.51i)12-s + (−5.33 + 2.21i)13-s + (3.72 − 0.367i)14-s − 3.58·15-s + (1.56 + 3.67i)16-s + (−0.0849 + 0.147i)17-s + ⋯ |
L(s) = 1 | + (0.287 − 0.957i)2-s + (0.0753 − 0.572i)3-s + (−0.834 − 0.551i)4-s + (−0.209 − 1.58i)5-s + (−0.526 − 0.236i)6-s + (0.380 + 0.924i)7-s + (−0.767 + 0.640i)8-s + (−0.321 − 0.0862i)9-s + (−1.58 − 0.257i)10-s + (−0.497 − 0.648i)11-s + (−0.378 + 0.436i)12-s + (−1.48 + 0.613i)13-s + (0.995 − 0.0981i)14-s − 0.925·15-s + (0.392 + 0.919i)16-s + (−0.0206 + 0.0356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.417863 + 0.891048i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.417863 + 0.891048i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.406 + 1.35i)T \) |
| 3 | \( 1 + (-0.130 + 0.991i)T \) |
| 7 | \( 1 + (-1.00 - 2.44i)T \) |
good | 5 | \( 1 + (0.467 + 3.55i)T + (-4.82 + 1.29i)T^{2} \) |
| 11 | \( 1 + (1.65 + 2.15i)T + (-2.84 + 10.6i)T^{2} \) |
| 13 | \( 1 + (5.33 - 2.21i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (0.0849 - 0.147i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 3.03i)T + (-4.91 - 18.3i)T^{2} \) |
| 23 | \( 1 + (-1.69 + 6.31i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (0.0420 - 0.0174i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (1.90 - 3.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.97 + 0.786i)T + (35.7 - 9.57i)T^{2} \) |
| 41 | \( 1 + (0.123 + 0.123i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.84 + 6.86i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (7.36 - 4.24i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.02 + 2.32i)T + (13.7 - 51.1i)T^{2} \) |
| 59 | \( 1 + (3.47 + 4.53i)T + (-15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (2.52 - 3.29i)T + (-15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (13.1 + 1.73i)T + (64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (-2.81 + 2.81i)T - 71iT^{2} \) |
| 73 | \( 1 + (-13.0 + 3.50i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.55 + 2.68i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.53 + 13.3i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (10.9 + 2.92i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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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
−9.849695215405619500415084448949, −8.927704361697222840341071969494, −8.644880977209287697547860155284, −7.57433313030983506926165066729, −6.00072955748332576419763603868, −5.00376290726670682056699561334, −4.63657749043031085266628346687, −2.89776121068735025156343881568, −1.88483383548441461060635502753, −0.46237122638017447666641870333,
2.77079937850432107920187650727, 3.69057411759913089196109476905, 4.70882980051665523089723978389, 5.64688651417517731618168112774, 6.85958168414405072530588820443, 7.59636535323112620418899884153, 7.84742271759485789799016231903, 9.667401702560831571477719312850, 9.946965333414865291343928302735, 10.91176587071628061254125999709