L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1 − 1.73i)5-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−0.999 − 1.73i)10-s + (−0.5 − 0.866i)11-s − 7·13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s − 1.99·20-s − 0.999·22-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (−3.5 + 6.06i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s − 1.94·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s − 0.447·20-s − 0.213·22-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.686 + 1.18i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.347146940890400031167254519453, −8.402711167269090113813159075923, −7.38053985947741429936379499473, −6.35447981703526983183437500642, −5.43014031734927876090125655462, −4.91386442592581204627499716929, −3.68612678543057485420691162952, −2.73874047689079527364115532002, −1.67638202708494261981625072922, 0,
2.41477780911271363056326392541, 3.06702934538345590599517576196, 4.34586621938657870757983730193, 5.15038466116120683249202141835, 6.22151945108275359559471990606, 6.87195576413003465722387076695, 7.35142333821652578905366075368, 8.393748599256950342727763959778, 9.463651737599295965675675668980