L(s) = 1 | − 0.530·5-s + i·7-s − 0.905i·11-s − 4.38i·13-s − 1.25i·17-s − 0.00364·19-s − 0.778·23-s − 4.71·25-s + 5.43·29-s + 3.49i·31-s − 0.530i·35-s − 5.34i·37-s + 1.74i·41-s + 0.259·43-s + 3.75·47-s + ⋯ |
L(s) = 1 | − 0.237·5-s + 0.377i·7-s − 0.272i·11-s − 1.21i·13-s − 0.304i·17-s − 0.000837·19-s − 0.162·23-s − 0.943·25-s + 1.00·29-s + 0.628i·31-s − 0.0895i·35-s − 0.878i·37-s + 0.272i·41-s + 0.0395·43-s + 0.548·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5078945075\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5078945075\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 0.530T + 5T^{2} \) |
| 11 | \( 1 + 0.905iT - 11T^{2} \) |
| 13 | \( 1 + 4.38iT - 13T^{2} \) |
| 17 | \( 1 + 1.25iT - 17T^{2} \) |
| 19 | \( 1 + 0.00364T + 19T^{2} \) |
| 23 | \( 1 + 0.778T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - 3.49iT - 31T^{2} \) |
| 37 | \( 1 + 5.34iT - 37T^{2} \) |
| 41 | \( 1 - 1.74iT - 41T^{2} \) |
| 43 | \( 1 - 0.259T + 43T^{2} \) |
| 47 | \( 1 - 3.75T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 - 5.73iT - 59T^{2} \) |
| 61 | \( 1 - 0.555iT - 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.96iT - 79T^{2} \) |
| 83 | \( 1 + 5.82iT - 83T^{2} \) |
| 89 | \( 1 + 1.94iT - 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.66180824007601582782940512820, −7.28510511715512386113692443670, −6.14069831943445985433205796346, −5.74447490777372780993664565599, −4.90809625334516171921343225051, −4.09465335906014947334258718164, −3.16788032200858322029506373491, −2.56031141544875883552891265246, −1.32146007378122249643601002097, −0.13230541179132730607373874619,
1.30567106949656293449875793487, 2.19777440012886365387940156176, 3.21564224636961907834829065413, 4.22991890641166154426263118480, 4.46337132277831322069424058748, 5.60214854996155587308609467201, 6.32895152578841529058316440838, 6.99964218793579890938787117647, 7.62657007105792623940890218194, 8.359501133922020662618300794079