L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.5 + 2.59i)3-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s + (−1.5 − 2.59i)6-s − 3·7-s + 0.999·8-s + (−3 − 5.19i)9-s + (−0.999 − 1.73i)10-s − 2·11-s + 3·12-s + (1.5 + 2.59i)13-s + (1.5 − 2.59i)14-s + (−3 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.866 + 1.49i)3-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + (−0.612 − 1.06i)6-s − 1.13·7-s + 0.353·8-s + (−1 − 1.73i)9-s + (−0.316 − 0.547i)10-s − 0.603·11-s + 0.866·12-s + (0.416 + 0.720i)13-s + (0.400 − 0.694i)14-s + (−0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (0.121 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.5 + 12.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.44352285086300015420734139080, −9.518580032017582096129309699952, −8.980756612361271874490696018293, −7.63670174069461568158016305560, −6.56714984365498803193934831293, −6.04372917736502442149991096708, −4.95124356927901601576876201963, −3.99209228214361539135954651834, −3.06142050202212257141506903621, 0,
1.02823271021404005042513044812, 2.44429873639467994954685809362, 3.77767311375806387082244958360, 5.30938132427389845527658135763, 6.03608112816006667343922363341, 7.09992574308365863368708802556, 7.86797887636509858048579923304, 8.567713903351625474916022369204, 9.722554163870129623924253296631