L(s) = 1 | + 0.385i·2-s + (1.56 − 0.742i)3-s + 1.85·4-s + 5-s + (0.286 + 0.603i)6-s + 1.48i·8-s + (1.89 − 2.32i)9-s + 0.385i·10-s − 2.54i·11-s + (2.89 − 1.37i)12-s − 3.06i·13-s + (1.56 − 0.742i)15-s + 3.12·16-s − 6.46·17-s + (0.896 + 0.731i)18-s − 1.19i·19-s + ⋯ |
L(s) = 1 | + 0.272i·2-s + (0.903 − 0.428i)3-s + 0.925·4-s + 0.447·5-s + (0.116 + 0.246i)6-s + 0.525i·8-s + (0.632 − 0.774i)9-s + 0.121i·10-s − 0.766i·11-s + (0.836 − 0.396i)12-s − 0.850i·13-s + (0.404 − 0.191i)15-s + 0.782·16-s − 1.56·17-s + (0.211 + 0.172i)18-s − 0.274i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.66506 - 0.362691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.66506 - 0.362691i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.56 + 0.742i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.385iT - 2T^{2} \) |
| 11 | \( 1 + 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 3.06iT - 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 1.19iT - 19T^{2} \) |
| 23 | \( 1 - 3.05iT - 23T^{2} \) |
| 29 | \( 1 - 7.77iT - 29T^{2} \) |
| 31 | \( 1 - 6.87iT - 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 + 9.34iT - 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 0.921iT - 71T^{2} \) |
| 73 | \( 1 + 0.296iT - 73T^{2} \) |
| 79 | \( 1 - 8.29T + 79T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 - 18.8T + 89T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−10.50942763221436089528682128684, −9.203996305859508553591928424639, −8.609318422628277716293956965055, −7.69320284265363860451378096504, −6.87348797025503953247056631989, −6.19787751797674601298504250804, −5.08149338862684713160346306139, −3.43512002161642410788794059295, −2.64298805089011233249058272186, −1.48045340051802795706594012770,
1.97044911053977487771165073377, 2.41935722358526563654089335616, 3.85895862231638651217171724676, 4.69174910463677022044188097278, 6.19593762351343304343250163525, 6.95039726079343459389425159605, 7.83928139832387695829710984634, 8.872374223672475488262652794734, 9.659026691546758505104128759557, 10.29571954089768530426858876493