L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.866 + 0.5i)3-s + (0.866 + 0.499i)4-s + (0.781 + 0.781i)5-s + (0.965 − 0.258i)6-s + (−2.22 − 1.43i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.552 − 0.957i)10-s + (0.312 − 1.16i)11-s − 12-s + (−2.62 + 2.47i)13-s + (1.77 + 1.96i)14-s + (−1.06 − 0.286i)15-s + (0.500 + 0.866i)16-s + (−1.19 + 2.06i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 + 0.249i)4-s + (0.349 + 0.349i)5-s + (0.394 − 0.105i)6-s + (−0.839 − 0.543i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.174 − 0.302i)10-s + (0.0943 − 0.352i)11-s − 0.288·12-s + (−0.728 + 0.685i)13-s + (0.474 + 0.524i)14-s + (−0.275 − 0.0739i)15-s + (0.125 + 0.216i)16-s + (−0.289 + 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0280954 - 0.132828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0280954 - 0.132828i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (2.22 + 1.43i)T \) |
| 13 | \( 1 + (2.62 - 2.47i)T \) |
good | 5 | \( 1 + (-0.781 - 0.781i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.312 + 1.16i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.19 - 2.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.87 - 1.30i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-5.04 + 2.91i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.18 + 7.25i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.07 + 6.07i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.63 - 9.84i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.0706 - 0.263i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (9.39 + 5.42i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.79 - 4.79i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + (2.46 + 9.18i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (9.83 + 5.67i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.76 + 1.54i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.24 + 8.36i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (4.16 - 4.16i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 + (-2.18 - 2.18i)T + 83iT^{2} \) |
| 89 | \( 1 + (-15.1 - 4.06i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.77 + 0.476i)T + (84.0 - 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.35520055381580338486797229199, −9.711784434096972814629311327200, −8.896897576667875786076917854975, −7.73148270425042037192303619596, −6.56514516852990116735363472945, −6.26082368694347259204746626493, −4.63437821355306744325567841165, −3.50972883978099256884067504557, −2.10721160541887437629425409652, −0.097277349342785129348372022874,
1.76844475447000647359003423269, 3.14698893584945079577813745501, 5.03246799663681675727614245734, 5.68803086934189427319001401233, 6.88191914284513605276380801338, 7.35593851601964016834061345165, 8.874741853399241819284777517554, 9.207719515589544231169636879036, 10.27759674736253206862034744205, 10.99115052694952947264493976038