L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 2·5-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (1 + 1.73i)10-s + 5·11-s + (−3 − 5.19i)13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (2 + 3.46i)17-s + (2 − 3.46i)19-s + (0.999 − 1.73i)20-s + (−2.5 − 4.33i)22-s − 4·23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (0.316 + 0.547i)10-s + 1.50·11-s + (−0.832 − 1.44i)13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.485 + 0.840i)17-s + (0.458 − 0.794i)19-s + (0.223 − 0.387i)20-s + (−0.533 − 0.923i)22-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2520128454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2520128454\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.5 + 6.06i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.04606473037796190443431056070, −9.359007501855332417732312408873, −8.466673374217054732087171258159, −7.62170484874550815547873658847, −6.92839724795108304249058462913, −5.86238034004201170196952722107, −4.56304648725065427304245471951, −3.58136113035547435633877093747, −3.06696002870834234948143484948, −1.27479362715653477529883469767,
0.13551756824583556318855705796, 1.97983036412156696180173622948, 3.67191903264812648699157957049, 4.18837361274079859330211458564, 5.54801338533978629810346687250, 6.42252269960447687974564183751, 7.15881380551124676361636732726, 7.77367562378114432082573297937, 8.950725681058421085899915573221, 9.448092749990224519202797238106