L(s) = 1 | − 1.73i·3-s + (0.5 − 0.866i)5-s + (−1.5 − 2.59i)7-s − 2.99·9-s + (2.5 + 4.33i)11-s + (2.5 − 4.33i)13-s + (−1.49 − 0.866i)15-s − 2·17-s + 4·19-s + (−4.5 + 2.59i)21-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s + 5.19i·27-s + (4.5 + 7.79i)29-s + (−0.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (0.223 − 0.387i)5-s + (−0.566 − 0.981i)7-s − 0.999·9-s + (0.753 + 1.30i)11-s + (0.693 − 1.20i)13-s + (−0.387 − 0.223i)15-s − 0.485·17-s + 0.917·19-s + (−0.981 + 0.566i)21-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s + 0.999i·27-s + (0.835 + 1.44i)29-s + (−0.0898 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.834050 - 0.699851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.834050 - 0.699851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.91711883807172063042696979752, −12.17827624041516065345586035141, −10.89080209585435461437045072888, −9.780415696448317845299387661805, −8.610030166282449061877628348495, −7.33291629688830953280640226187, −6.64745874278211041824038420645, −5.17698809392735266441884198534, −3.39625860626815687212383917190, −1.31259399580731222414321464551,
2.83772857287000543069694881152, 4.10744132624351610669430925227, 5.76344677891560334785169615219, 6.49326918264950109464074853900, 8.608424029237451573602788670117, 9.110252047570485475361892622153, 10.20837879914991791089988814963, 11.37394362891619155388960776176, 11.94210221340455455198242576996, 13.69769288080972841376805078173