L(s) = 1 | + (1.32 + 0.480i)2-s + (−1.31 − 1.12i)3-s + (1.53 + 1.27i)4-s + (1.86 − 3.22i)5-s + (−1.20 − 2.13i)6-s + (−2.46 − 0.962i)7-s + (1.43 + 2.44i)8-s + (0.465 + 2.96i)9-s + (4.02 − 3.39i)10-s + (2.26 − 1.31i)11-s + (−0.584 − 3.41i)12-s + 3.57i·13-s + (−2.81 − 2.46i)14-s + (−6.07 + 2.14i)15-s + (0.729 + 3.93i)16-s + (−0.186 + 0.107i)17-s + ⋯ |
L(s) = 1 | + (0.940 + 0.339i)2-s + (−0.760 − 0.649i)3-s + (0.768 + 0.639i)4-s + (0.832 − 1.44i)5-s + (−0.493 − 0.869i)6-s + (−0.931 − 0.363i)7-s + (0.505 + 0.862i)8-s + (0.155 + 0.987i)9-s + (1.27 − 1.07i)10-s + (0.684 − 0.395i)11-s + (−0.168 − 0.985i)12-s + 0.990i·13-s + (−0.752 − 0.658i)14-s + (−1.56 + 0.554i)15-s + (0.182 + 0.983i)16-s + (−0.0451 + 0.0260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59200 - 0.395001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59200 - 0.395001i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.32 - 0.480i)T \) |
| 3 | \( 1 + (1.31 + 1.12i)T \) |
| 7 | \( 1 + (2.46 + 0.962i)T \) |
good | 5 | \( 1 + (-1.86 + 3.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.26 + 1.31i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.57iT - 13T^{2} \) |
| 17 | \( 1 + (0.186 - 0.107i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.14 - 1.97i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.33 - 4.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 + (4.26 - 2.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.31 + 4.22i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.909iT - 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + (0.586 - 1.01i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.06 + 1.83i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.79 + 3.92i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.301 - 0.173i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + (-4.45 - 7.71i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.70 - 5.60i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.73iT - 83T^{2} \) |
| 89 | \( 1 + (15.4 + 8.93i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.61344370662940618348279981026, −12.28884922612593979685272486677, −11.10487239072162611098836426919, −9.671654078335908161051278213168, −8.494396155773649427447997803039, −7.02186624410953954311831669368, −6.14438492337106060967939773340, −5.31094657490761751264794019209, −4.03765172517729952932813072469, −1.70771247372995205106011444245,
2.63913785837073377663798864332, 3.77104268541213707809565127991, 5.40394982530367621635167047112, 6.33357032063199911161423096072, 6.87661745399921300233181208551, 9.432853044478742984186255192400, 10.26454078800197126769109459054, 10.75841332839223646265343357179, 11.91110363395104464244328785707, 12.76961904074527533967702372348