L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + 3.05i·5-s + (0.866 + 0.499i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.52 + 2.64i)10-s + (−2.98 + 1.72i)11-s + 0.999·12-s + (3.25 − 1.55i)13-s + 0.999·14-s + (−2.64 + 1.52i)15-s + (−0.5 − 0.866i)16-s + (−1.41 + 2.44i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + 1.36i·5-s + (0.353 + 0.204i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.483 + 0.836i)10-s + (−0.898 + 0.518i)11-s + 0.288·12-s + (0.902 − 0.431i)13-s + 0.267·14-s + (−0.683 + 0.394i)15-s + (−0.125 − 0.216i)16-s + (−0.343 + 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95825 + 1.01573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95825 + 1.01573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-3.25 + 1.55i)T \) |
good | 5 | \( 1 - 3.05iT - 5T^{2} \) |
| 11 | \( 1 + (2.98 - 1.72i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.41 - 2.44i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.49 - 0.862i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.92 + 3.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.978iT - 31T^{2} \) |
| 37 | \( 1 + (-4.58 + 2.64i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.62 + 4.98i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.51 - 2.61i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8.04iT - 47T^{2} \) |
| 53 | \( 1 + 8.33T + 53T^{2} \) |
| 59 | \( 1 + (-8.64 - 4.98i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.77 + 9.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.11 + 2.37i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.17 + 1.83i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 4.49T + 79T^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 + (6.84 - 3.95i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.88 + 4.54i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.89294340653573189869796428081, −10.36055270479569596047627452523, −9.459399000979135741369637108894, −8.167924664204393500307798341725, −7.31572537034515420682385259354, −6.18486553817925258198178790297, −5.31198564109933182563469889970, −4.03818256803755228264191672854, −3.13249927276825195201384804925, −2.15183689530463916303899119836,
1.11541997439468983081592657501, 2.73310824370063157662183896347, 4.16483731699570470533197513691, 5.04892507829891282149478440926, 5.92160382064144849344900282171, 7.06573337410252472382046795962, 8.086954467108983644981362737650, 8.606839642397001830075488635440, 9.502790436806801175032102866839, 11.01241482286716819407308027184