L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s − 1.73i·6-s + 0.999·8-s + (1.5 + 2.59i)9-s − 3·10-s + (1.5 + 2.59i)11-s + (−1.49 + 0.866i)12-s + (−2.5 + 4.33i)13-s + (4.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + 3·17-s + (1.5 − 2.59i)18-s + 5·19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s − 0.707i·6-s + 0.353·8-s + (0.5 + 0.866i)9-s − 0.948·10-s + (0.452 + 0.783i)11-s + (−0.433 + 0.250i)12-s + (−0.693 + 1.20i)13-s + (1.16 − 0.670i)15-s + (−0.125 − 0.216i)16-s + 0.727·17-s + (0.353 − 0.612i)18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99427 - 0.351644i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99427 - 0.351644i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 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 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.858940463072466950650768035008, −9.185203477159087046309975393721, −8.932814381194564121741436678409, −7.78640788603460814807806284943, −6.94367835896478384319121548641, −5.27307644448224853393787628935, −4.64981015163431147105152991101, −3.66251987396276502763379907594, −2.30747683987178231865979425522, −1.42417440618217069832351905031,
1.23848234007064724663287252461, 2.81547609248212821861227976836, 3.36968975924260278622053546074, 5.18138152338355348879699508332, 6.15557701072660678744916743202, 6.83249644316601599513517718260, 7.71200222627466327012996492541, 8.230916528597595630748826748176, 9.490487128299676456524541004090, 9.833969949560615810373674139956