L(s) = 1 | − 2.52i·5-s − 7-s − 5.70i·11-s − 3.06i·13-s − 5.49·17-s − 7.28i·19-s − 0.539·23-s − 1.38·25-s + 8.35i·29-s − 6.74·31-s + 2.52i·35-s − 10.2i·37-s + 5.58·41-s − 3.98i·43-s − 4.83·47-s + ⋯ |
L(s) = 1 | − 1.12i·5-s − 0.377·7-s − 1.71i·11-s − 0.848i·13-s − 1.33·17-s − 1.67i·19-s − 0.112·23-s − 0.276·25-s + 1.55i·29-s − 1.21·31-s + 0.427i·35-s − 1.68i·37-s + 0.872·41-s − 0.607i·43-s − 0.705·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9046843633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9046843633\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 5 | \( 1 + 2.52iT - 5T^{2} \) |
| 11 | \( 1 + 5.70iT - 11T^{2} \) |
| 13 | \( 1 + 3.06iT - 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 + 7.28iT - 19T^{2} \) |
| 23 | \( 1 + 0.539T + 23T^{2} \) |
| 29 | \( 1 - 8.35iT - 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 + 3.98iT - 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 + 4.14iT - 61T^{2} \) |
| 67 | \( 1 - 5.96iT - 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 5.83iT - 83T^{2} \) |
| 89 | \( 1 - 9.28T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.73115987410144979345289144388, −6.92390117893546530271851248385, −6.16373734460519624817197589826, −5.38169003694391935724098759402, −4.94495160353685830388634687501, −3.93333572331787940377327364558, −3.19003437145482329086385788313, −2.28471906202564851448548023290, −0.920788483595561710583194710915, −0.26331861170681083869672527679,
1.79002229191483692473834647574, 2.26452577023267493725061747598, 3.30484355422941595711742436320, 4.17225811552678532364426657595, 4.66650413071438039748119202421, 5.89444831002558100862313839490, 6.50578345092676767609096928707, 7.00417708455286046845743024809, 7.61085091631226768155156237099, 8.400510041153483937116683419090