L(s) = 1 | + (−0.866 + 0.5i)2-s + (−1.47 + 0.851i)3-s + (0.499 − 0.866i)4-s + (0.851 − 1.47i)6-s + 3.74i·7-s + 0.999i·8-s + (−0.0492 + 0.0852i)9-s + 3.64·11-s + 1.70i·12-s + (5.23 + 3.01i)13-s + (−1.87 − 3.24i)14-s + (−0.5 − 0.866i)16-s + (3.54 − 2.04i)17-s − 0.0984i·18-s + (0.697 − 4.30i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.851 + 0.491i)3-s + (0.249 − 0.433i)4-s + (0.347 − 0.602i)6-s + 1.41i·7-s + 0.353i·8-s + (−0.0164 + 0.0284i)9-s + 1.09·11-s + 0.491i·12-s + (1.45 + 0.837i)13-s + (−0.500 − 0.866i)14-s + (−0.125 − 0.216i)16-s + (0.860 − 0.497i)17-s − 0.0231i·18-s + (0.160 − 0.987i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.426414 + 0.821962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.426414 + 0.821962i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + (-0.697 + 4.30i)T \) |
good | 3 | \( 1 + (1.47 - 0.851i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 3.74iT - 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + (-5.23 - 3.01i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.54 + 2.04i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.05 + 2.34i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.32 - 5.75i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 7.75iT - 37T^{2} \) |
| 41 | \( 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.80 - 2.19i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.50 - 0.871i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.50 - 0.871i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.67 + 6.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.15 - 1.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.83 + 3.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.994 - 1.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.95 - 4.59i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.07 - 5.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.69iT - 83T^{2} \) |
| 89 | \( 1 + (5.53 - 9.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.30 + 0.752i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.21407558822704510524853960258, −9.399968394658265215352467678711, −8.774573983994419276884984363751, −8.063196181559678405937424836188, −6.54384746721397863941672572589, −6.22148005320612470960297412355, −5.29327823177135352619056748076, −4.38527203018845073010405083734, −2.86678153780234802492758384624, −1.35248002451110378295912532316,
0.72826884029468695304364096955, 1.45244346206368166916246875120, 3.55063028832187021057856011319, 4.01298717817309907462887713989, 5.86112441808051723485916493111, 6.22123941718303106320315540399, 7.34351506436602177716673338366, 7.953102674501868814722920438061, 8.932786155201019152056665882194, 10.08813021749990395065873646594