L(s) = 1 | − i·3-s + 0.585i·5-s − 3.41·7-s − 9-s + 2i·11-s − 2.82i·13-s + 0.585·15-s + 3.65·17-s + 5.65i·19-s + 3.41i·21-s − 1.17·23-s + 4.65·25-s + i·27-s − 0.585i·29-s + 4.58·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.261i·5-s − 1.29·7-s − 0.333·9-s + 0.603i·11-s − 0.784i·13-s + 0.151·15-s + 0.886·17-s + 1.29i·19-s + 0.745i·21-s − 0.244·23-s + 0.931·25-s + 0.192i·27-s − 0.108i·29-s + 0.823·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.376685892\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.376685892\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 - 0.585iT - 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 0.585iT - 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 9.65iT - 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 1.65iT - 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.8iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 9.65iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 9.17T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 - 5.75T + 79T^{2} \) |
| 83 | \( 1 + 9.31iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.534766737856858134885186883381, −8.654921430170131458752864256980, −7.56926843753550460606630417946, −7.24565716450861065065094149492, −5.99147508917696044031915295733, −5.81349288483746806590914092557, −4.26759341834487760253298501176, −3.26220628638032236354162177701, −2.46685874797301193814356934110, −0.919227192973046285345743900194,
0.74591441958885828770525285792, 2.64097456115092330237852405054, 3.39775947061958694993848936619, 4.38208042267519582795814839890, 5.28032299262245179452380244487, 6.27109582918201779910648328658, 6.83579435721434827338561810681, 7.971820089118118289789832604895, 8.951955517105535879197790904475, 9.362343567005639172030083570546