L(s) = 1 | − 1.73i·3-s + (1.5 − 0.866i)5-s + (−1.5 − 0.866i)7-s − 2.99·9-s + (1.5 − 2.59i)11-s + (2.5 + 4.33i)13-s + (−1.49 − 2.59i)15-s − 6.92i·17-s + 3.46i·19-s + (−1.49 + 2.59i)21-s + (4.5 + 7.79i)23-s + (−1 + 1.73i)25-s + 5.19i·27-s + (1.5 + 0.866i)29-s + (−4.5 + 2.59i)31-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (0.670 − 0.387i)5-s + (−0.566 − 0.327i)7-s − 0.999·9-s + (0.452 − 0.783i)11-s + (0.693 + 1.20i)13-s + (−0.387 − 0.670i)15-s − 1.68i·17-s + 0.794i·19-s + (−0.327 + 0.566i)21-s + (0.938 + 1.62i)23-s + (−0.200 + 0.346i)25-s + 0.999i·27-s + (0.278 + 0.160i)29-s + (−0.808 + 0.466i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.941878 - 0.659510i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.941878 - 0.659510i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \) |
| 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 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (7.5 + 4.33i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.22180926602360114803102296989, −11.87905048699547129785020985695, −11.20291703544438106782868112281, −9.511677227347317335941811743246, −8.885870633594408685217822171037, −7.39513206385297268528046215994, −6.46462364300244324935226542980, −5.39849902506889789807652657025, −3.35383496393844425103258497966, −1.43834980079011542247287906633,
2.71520314460127089964457787895, 4.14480418184802737496489738642, 5.65049118824998910766678737467, 6.53807913591060777441140676272, 8.361474323775439297448556447842, 9.312944172572840639402570621401, 10.34028644547091426521377671830, 10.84736513457373417686869429970, 12.40152109552675379619288166045, 13.24115208490608309117435303920