L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s − 4·13-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (−1 + 1.73i)19-s + (2.5 + 4.33i)25-s + (2 − 3.46i)26-s + 6·29-s + (2 + 3.46i)31-s + (−0.499 − 0.866i)32-s − 6·34-s + (−1 + 1.73i)37-s + (−0.999 − 1.73i)38-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.353·8-s − 1.10·13-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (−0.229 + 0.397i)19-s + (0.5 + 0.866i)25-s + (0.392 − 0.679i)26-s + 1.11·29-s + (0.359 + 0.622i)31-s + (−0.0883 − 0.153i)32-s − 1.02·34-s + (−0.164 + 0.284i)37-s + (−0.162 − 0.280i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.554939 + 0.834266i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.554939 + 0.834266i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.24050168938634619756520511423, −9.562839283642452873150273824419, −8.579070835181114491590579675972, −7.893845413274031277357094778620, −7.04484775800697037631545065049, −6.16705628437093657687964839083, −5.25382606869139825437544321754, −4.30707667531637213604764354311, −2.96914043439670374576015188211, −1.41601001243632920313198798025,
0.57480985419816511141810679089, 2.27167569372610041124362855296, 3.13792778408072637693602168647, 4.48973455666388034502967389192, 5.22717740194178482115636502177, 6.61190330864623345980003208225, 7.43214040408875706848220213907, 8.290591895518752341408238230342, 9.202208327398030096372454610694, 9.947962399167162574763516562225