L(s) = 1 | + 2-s − 4-s + 2i·5-s + 4i·7-s − 3·8-s + 3·9-s + 2i·10-s − 2·13-s + 4i·14-s − 16-s + 3·18-s + 4·19-s − 2i·20-s + 4i·23-s + 25-s − 2·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.5·4-s + 0.894i·5-s + 1.51i·7-s − 1.06·8-s + 9-s + 0.632i·10-s − 0.554·13-s + 1.06i·14-s − 0.250·16-s + 0.707·18-s + 0.917·19-s − 0.447i·20-s + 0.834i·23-s + 0.200·25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18300 + 0.923664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18300 + 0.923664i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−12.11576759312387413270668491299, −11.38819580280739913467497946584, −9.895173764373709135343902008222, −9.404339157686116863184330181939, −8.151160348286873343769939957815, −6.93489966522597287012637266133, −5.82753729752985931996525656458, −4.96451134545497643237637350602, −3.61419840993050833045566105447, −2.44234076885814509524119187057,
1.01140502692843763436254945597, 3.48509309417351538944822117777, 4.53772610926885161211825836280, 5.04654405096225976917833244370, 6.67831097603677793979582836253, 7.62628132785994960680224184756, 8.822103400278922359567134491236, 9.785764555801647966760879556209, 10.55060498663064660633869541862, 11.96576197067687149047187743735