| L(s) = 1 | − 2.04i·5-s − 7.79·7-s − 4.09i·11-s − 6.69·13-s + 19.3i·17-s − 1.79·19-s − 14.1i·23-s + 20.7·25-s + 28.4i·29-s − 20.2·31-s + 15.9i·35-s + 41.5·37-s − 4.94i·41-s + 75.1·43-s + 13.5i·47-s + ⋯ |
| L(s) = 1 | − 0.409i·5-s − 1.11·7-s − 0.372i·11-s − 0.515·13-s + 1.13i·17-s − 0.0946·19-s − 0.614i·23-s + 0.831·25-s + 0.982i·29-s − 0.651·31-s + 0.456i·35-s + 1.12·37-s − 0.120i·41-s + 1.74·43-s + 0.288i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.320747117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.320747117\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.04iT - 25T^{2} \) |
| 7 | \( 1 + 7.79T + 49T^{2} \) |
| 11 | \( 1 + 4.09iT - 121T^{2} \) |
| 13 | \( 1 + 6.69T + 169T^{2} \) |
| 17 | \( 1 - 19.3iT - 289T^{2} \) |
| 19 | \( 1 + 1.79T + 361T^{2} \) |
| 23 | \( 1 + 14.1iT - 529T^{2} \) |
| 29 | \( 1 - 28.4iT - 841T^{2} \) |
| 31 | \( 1 + 20.2T + 961T^{2} \) |
| 37 | \( 1 - 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 4.94iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 75.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 13.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 54.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 89.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 37.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 48.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 92.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 158. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 121. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 167.T + 9.40e3T^{2} \) |
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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
−9.578382053168253335205338607883, −8.958804626163893829890855215673, −8.144932175352353169393086485142, −7.11669813751683413135804912012, −6.32141393137699887234250788874, −5.53939673284884446380291248129, −4.42405691501680652272081295142, −3.47579527182586475759154831883, −2.43456415697479179377178995092, −0.881610042346681328581967062121,
0.52246481841052662671009862019, 2.33972441163609215973942932306, 3.14721144292268257237467614207, 4.22873558853326327600356486168, 5.31059037399401133859376336676, 6.26957457675109270341814511352, 7.06446743186970534935200489451, 7.65063203582367473255768267030, 8.906526700798962619305965504862, 9.662685381118184335414622301854
