L(s) = 1 | + (1.5 + 0.866i)2-s + (1.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (1.5 + 2.59i)6-s + (2.5 + 0.866i)7-s − 1.73i·8-s + (1.5 + 2.59i)9-s + (−3 + 1.73i)11-s + 1.73i·12-s − 3.46i·13-s + (3 + 3.46i)14-s + (2.49 − 4.33i)16-s + (3 + 5.19i)17-s + 5.19i·18-s + (−6 − 3.46i)19-s + ⋯ |
L(s) = 1 | + (1.06 + 0.612i)2-s + (0.866 + 0.499i)3-s + (0.250 + 0.433i)4-s + (0.612 + 1.06i)6-s + (0.944 + 0.327i)7-s − 0.612i·8-s + (0.5 + 0.866i)9-s + (−0.904 + 0.522i)11-s + 0.500i·12-s − 0.960i·13-s + (0.801 + 0.925i)14-s + (0.624 − 1.08i)16-s + (0.727 + 1.26i)17-s + 1.22i·18-s + (−1.37 − 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.76121 + 1.71306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.76121 + 1.71306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6 + 3.46i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−10.70127890049395964567527193255, −10.34251441744039307663699741978, −9.102024485599668181579034170595, −8.080544864301095215912213665398, −7.58257251726555832334248093779, −6.16504152079227539023903952816, −5.16099776929738346966131980544, −4.53253895363023009489259690798, −3.45618207175185937768537404501, −2.13714274851375833883941155088,
1.72125123585601701556803760529, 2.73364077566756606540380404057, 3.84579089588615815391573623526, 4.71982483392837618333820171144, 5.82777083181418105434732987039, 7.19067669217010053100632702220, 8.077504663606922395348164915554, 8.693219895861076343917781130210, 9.976782393633937247258553713503, 11.00654504979052334492958365712