L(s) = 1 | + (0.5 − 0.866i)2-s + (1.68 + 0.396i)3-s + (−0.499 − 0.866i)4-s + (1.5 + 1.65i)5-s + (1.18 − 1.26i)6-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (2.68 + 1.33i)9-s + (2.18 − 0.469i)10-s + (3.68 − 2.12i)11-s + (−0.499 − 1.65i)12-s − 2·13-s + (−2 + 1.73i)14-s + (1.87 + 3.39i)15-s + (−0.5 + 0.866i)16-s + (−5.74 + 3.31i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.973 + 0.228i)3-s + (−0.249 − 0.433i)4-s + (0.670 + 0.741i)5-s + (0.484 − 0.515i)6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (0.895 + 0.445i)9-s + (0.691 − 0.148i)10-s + (1.11 − 0.641i)11-s + (−0.144 − 0.478i)12-s − 0.554·13-s + (−0.534 + 0.462i)14-s + (0.483 + 0.875i)15-s + (−0.125 + 0.216i)16-s + (−1.39 + 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81852 - 0.444314i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81852 - 0.444314i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.68 - 0.396i)T \) |
| 5 | \( 1 + (-1.5 - 1.65i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 11 | \( 1 + (-3.68 + 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (5.74 - 3.31i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.18 + 3.78i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.31iT - 29T^{2} \) |
| 31 | \( 1 + (2.05 - 1.18i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (10.1 + 5.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.62T + 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 + (-1.62 - 0.939i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.686 - 1.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.05 - 3.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.44 + 1.40i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.55 + 3.78i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.87iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.05 + 7.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.43iT - 83T^{2} \) |
| 89 | \( 1 + (-2.18 + 3.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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
−12.61523047903781057990057885189, −11.05709194681835258285927601693, −10.39443270870447638832615489860, −9.400744753877473690358494306498, −8.761332846228992904031992527398, −6.97982584962733298409023531441, −6.22878800057677037886620362661, −4.33961987630648039761604936268, −3.30524484863273526556356751601, −2.16440665677769000526595818414,
2.15190011609832083644186196865, 3.76191248847907247816191456230, 5.01276061802680401246450665826, 6.50667406881112696311230589551, 7.14233005846698791964978395761, 8.758011600264658810037086445088, 9.136779732602358153146252332528, 10.02057831083790530269383514224, 12.01770822666816090847721511723, 12.70418170993660781363266840811