L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·6-s + (0.844 + 1.46i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s − 3.20·11-s + (0.499 − 0.866i)12-s + (3.91 − 2.26i)13-s − 1.68i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (−2.32 − 1.34i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + 0.408i·6-s + (0.319 + 0.553i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s − 0.965·11-s + (0.144 − 0.249i)12-s + (1.08 − 0.627i)13-s − 0.451i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (−0.564 − 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09006156122\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09006156122\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (6.07 + 0.237i)T \) |
good | 7 | \( 1 + (-0.844 - 1.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 + (-3.91 + 2.26i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.32 + 1.34i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.29 - 1.32i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6.25iT - 23T^{2} \) |
| 29 | \( 1 + 6.29iT - 29T^{2} \) |
| 31 | \( 1 + 2.13iT - 31T^{2} \) |
| 41 | \( 1 + (5.65 + 9.78i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 4.12iT - 43T^{2} \) |
| 47 | \( 1 + 8.83T + 47T^{2} \) |
| 53 | \( 1 + (6.84 - 11.8i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.611 - 0.352i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.81 - 5.08i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.212 + 0.367i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.72 - 6.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + (-3.03 + 1.75i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.09 + 5.36i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.23 - 0.710i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−9.314541761686748656607519785637, −8.419063684369893702451256375991, −7.917756995767689714637399167116, −7.08916722717825470672484301238, −6.02970806870266608793679590777, −5.25509727735260744592247048427, −3.86624669243664661535376496438, −2.76142356100917998388575707758, −1.68824892454472151296082299957, −0.05015036711105207806542430430,
1.57262685467464723290722826836, 3.21146727522679967046008094904, 4.44777446852670194479202468922, 5.03469203859666922909858237877, 6.33454602962637800345643577983, 6.87622134463958837925955834138, 8.149915266956762559056680889733, 8.478504445522370168903502375725, 9.403730548201563412911010067674, 10.45942527591398934790215923138