| L(s) = 1 | + (1.36 + 0.366i)2-s + (2.92 − 0.677i)3-s + (1.73 + i)4-s + (4.99 − 0.0656i)5-s + (4.24 + 0.144i)6-s + (−2.70 + 6.45i)7-s + (1.99 + 2i)8-s + (8.08 − 3.95i)9-s + (6.85 + 1.74i)10-s + (−17.7 − 10.2i)11-s + (5.73 + 1.74i)12-s + (10.2 + 10.2i)13-s + (−6.06 + 7.82i)14-s + (14.5 − 3.57i)15-s + (1.99 + 3.46i)16-s + (−16.4 + 4.39i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.974 − 0.225i)3-s + (0.433 + 0.250i)4-s + (0.999 − 0.0131i)5-s + (0.706 + 0.0241i)6-s + (−0.386 + 0.922i)7-s + (0.249 + 0.250i)8-s + (0.898 − 0.439i)9-s + (0.685 + 0.174i)10-s + (−1.61 − 0.933i)11-s + (0.478 + 0.145i)12-s + (0.790 + 0.790i)13-s + (−0.432 + 0.559i)14-s + (0.971 − 0.238i)15-s + (0.124 + 0.216i)16-s + (−0.965 + 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.20698 + 0.423046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.20698 + 0.423046i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 3 | \( 1 + (-2.92 + 0.677i)T \) |
| 5 | \( 1 + (-4.99 + 0.0656i)T \) |
| 7 | \( 1 + (2.70 - 6.45i)T \) |
| good | 11 | \( 1 + (17.7 + 10.2i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-10.2 - 10.2i)T + 169iT^{2} \) |
| 17 | \( 1 + (16.4 - 4.39i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (11.5 + 20.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-5.50 + 20.5i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 2.30T + 841T^{2} \) |
| 31 | \( 1 + (-3.09 - 1.78i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (10.7 - 40.1i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 0.0268T + 1.68e3T^{2} \) |
| 43 | \( 1 + (6.00 - 6.00i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-3.18 + 11.8i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (8.99 - 2.41i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (13.7 + 7.94i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-41.9 + 24.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (23.7 - 6.37i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (137. - 36.9i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (22.7 - 13.1i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-83.7 + 83.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (39.3 - 22.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-84.5 + 84.5i)T - 9.40e3iT^{2} \) |
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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
−12.76670494725246628391148814192, −11.25170906394590906040343020663, −10.22128581953076888289537000110, −8.862342859847319668850532682553, −8.475682038138132777159386688596, −6.80447641244415615272321639925, −6.02737347690051134950592051072, −4.74304535088248849005094763992, −2.99874470324348386212584917255, −2.21129295192354186746657232560,
1.93211459821078053118787732574, 3.14101382081667813740743621486, 4.43686876699435431741992237869, 5.64552101317315710336968301560, 7.02524410757946639020979296124, 7.982661642719391406251957478159, 9.370388027753429953934991425901, 10.38769253168940732250647624677, 10.64386240528377221621370491718, 12.70700868199878046820012750499
