L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−1 + 1.73i)5-s + (2 + 3.46i)7-s − 3·8-s + 1.99·10-s + (−2 − 3.46i)11-s + (−0.5 + 0.866i)13-s + (1.99 − 3.46i)14-s + (0.500 + 0.866i)16-s + 2·17-s + (0.999 + 1.73i)20-s + (−1.99 + 3.46i)22-s + (0.500 + 0.866i)25-s + 0.999·26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.755 + 1.30i)7-s − 1.06·8-s + 0.632·10-s + (−0.603 − 1.04i)11-s + (−0.138 + 0.240i)13-s + (0.534 − 0.925i)14-s + (0.125 + 0.216i)16-s + 0.485·17-s + (0.223 + 0.387i)20-s + (−0.426 + 0.738i)22-s + (0.100 + 0.173i)25-s + 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.102250467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102250467\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6 - 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.25697650412863615088700128720, −9.096259295277297379348565894476, −8.599251330176785936745951053385, −7.63760159610305741771359912362, −6.55300868852639826595281104303, −5.70556292082506757176595982950, −4.97656881435697260778614980774, −3.21106786701521275019249777931, −2.69980677707273842439706379571, −1.40526707573869599306845730655,
0.57784854588684380191294648504, 2.26798651304069144346399973742, 3.80735421179063566397293340183, 4.52850970604890807957077009337, 5.50318358385417967926218792156, 6.81373758955691357679580602282, 7.52120848754557787243523229287, 7.961657803175276911099579320558, 8.663604449681589915151562912100, 9.799815301807426672559061395746