L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (1.81 − 1.30i)5-s + (0.499 − 0.866i)6-s + 3i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.917 − 2.03i)10-s + 2·11-s − 0.999i·12-s + (−2.98 − 1.72i)13-s + (1.5 + 2.59i)14-s + (0.917 − 2.03i)15-s + (−0.5 − 0.866i)16-s + (4.24 − 2.44i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.811 − 0.584i)5-s + (0.204 − 0.353i)6-s + 1.13i·7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.290 − 0.644i)10-s + 0.603·11-s − 0.288i·12-s + (−0.828 − 0.478i)13-s + (0.400 + 0.694i)14-s + (0.236 − 0.526i)15-s + (−0.125 − 0.216i)16-s + (1.02 − 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.43709 - 1.21946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.43709 - 1.21946i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.81 + 1.30i)T \) |
| 19 | \( 1 + (-1 - 4.24i)T \) |
good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (2.98 + 1.72i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.24 + 2.44i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.12 + 1.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.67 - 8.09i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 + 4.55iT - 37T^{2} \) |
| 41 | \( 1 + (1.22 + 2.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.476 - 0.275i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.07 + 1.77i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.68 + 1.55i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.89 - 10.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.17 - 3.76i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.25 + 0.724i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.77 - 3.07i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 5.94i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.39 - 14.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.34iT - 83T^{2} \) |
| 89 | \( 1 + (5.77 - 10.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (16.1 - 9.34i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.55033561017287013839251588737, −9.556536978264385390555111519514, −9.126813652574951004628257180170, −8.016781752210887923662683048998, −6.89487227797760278319270378838, −5.57767118328807218823483265469, −5.34473911115912376751137694792, −3.73134574320151464956804465359, −2.56330472061208517004571420226, −1.56037044057019675620847923533,
1.91163064807301219454627926060, 3.28370245451862360023301902396, 4.14612999618081273438881165490, 5.27384510322405095464558917918, 6.40012795227221723711742586031, 7.20414389002981757104704642571, 7.924179990473238162378085640023, 9.378934518195696829339352655042, 9.856401083719896385618587295005, 10.83782352612323220967088997603