L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s − i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999i·10-s + (−0.866 − 0.499i)12-s + (−0.5 + 0.866i)14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)21-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s − i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999i·10-s + (−0.866 − 0.499i)12-s + (−0.5 + 0.866i)14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2643675401\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2643675401\) |
\(L(1)\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.600885542656583058190950576172, −8.970593673519577403520914115601, −7.87445726773197395752230215783, −7.34183347441598324729591178624, −6.28857878887941272030148867303, −5.17900827712441822337566401955, −4.07132290572644014598031320381, −3.65380538191506123414068692340, −1.66805206461756306521985861252, −0.33659931466699805046730468808,
1.74038011533462481330257198059, 2.83421344021656924349063129716, 4.53137574473272320847030183592, 5.72373977554336991486304502272, 6.12947245837479072340110608020, 7.06656770211473777228857460989, 7.66225347360381498438974373800, 8.470227519839690676638207329310, 9.414050958597604973208807219225, 10.33849601085739142457970225840