L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.48 + 1.67i)5-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−0.448 − 2.19i)10-s + (2 + 3.46i)11-s − 4.24i·13-s + (−0.5 + 0.866i)16-s + (−3.67 + 2.12i)17-s + (2.59 − 1.5i)18-s + (−2.82 + 4.89i)19-s + (−0.707 + 2.12i)20-s − 3.99i·22-s + (−0.598 + 4.96i)25-s + (−2.12 + 3.67i)26-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.663 + 0.748i)5-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.141 − 0.692i)10-s + (0.603 + 1.04i)11-s − 1.17i·13-s + (−0.125 + 0.216i)16-s + (−0.891 + 0.514i)17-s + (0.612 − 0.353i)18-s + (−0.648 + 1.12i)19-s + (−0.158 + 0.474i)20-s − 0.852i·22-s + (−0.119 + 0.992i)25-s + (−0.416 + 0.720i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.856747 + 0.571421i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.856747 + 0.571421i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.48 - 1.67i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + (3.67 - 2.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (2.82 + 4.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.19 - 3i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.65 - 9.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.53 + 6.12i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-3.67 + 2.12i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.9iT - 83T^{2} \) |
| 89 | \( 1 + (-2.12 + 3.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.24iT - 97T^{2} \) |
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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
−10.86295572526166346269242938717, −10.27861576006669446163233013971, −9.586892172178479218445676642873, −8.422247769748360655522971248473, −7.66917839999462233189733964939, −6.58851643816266308161431668172, −5.69341275887557825447193770535, −4.24807527817287375369643201222, −2.79173113980320049660532182227, −1.84844258356382484747429837283,
0.76220086117549775617157546450, 2.37090541117440441318954254126, 4.08467674449296409002398588219, 5.31386649484653730912403501380, 6.37037403514766418830205341378, 6.88855535489482049312375509870, 8.601110976629843134000374342620, 8.908151022074489562386960957149, 9.492924757782086043864086324783, 10.79196698774209708067991739164