| L(s) = 1 | − 8.91i·2-s + (−18.6 + 42.8i)3-s + 48.5·4-s − 62.4·5-s + (382. + 166. i)6-s − 1.57e3i·8-s + (−1.48e3 − 1.60e3i)9-s + 556. i·10-s + 1.51e3i·11-s + (−906. + 2.08e3i)12-s + 1.53e3i·13-s + (1.16e3 − 2.67e3i)15-s − 7.81e3·16-s + 1.19e4·17-s + (−1.42e4 + 1.32e4i)18-s + 1.71e4i·19-s + ⋯ |
| L(s) = 1 | − 0.787i·2-s + (−0.399 + 0.916i)3-s + 0.379·4-s − 0.223·5-s + (0.722 + 0.314i)6-s − 1.08i·8-s + (−0.680 − 0.732i)9-s + 0.176i·10-s + 0.343i·11-s + (−0.151 + 0.347i)12-s + 0.193i·13-s + (0.0892 − 0.204i)15-s − 0.477·16-s + 0.590·17-s + (−0.577 + 0.536i)18-s + 0.572i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.06109935688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06109935688\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (18.6 - 42.8i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 8.91iT - 128T^{2} \) |
| 5 | \( 1 + 62.4T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.51e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.53e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.19e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.71e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 5.66e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 2.39e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.43e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 2.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.06e3T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.32e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.07e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.94e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.82e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 7.17e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.62e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 4.72e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 5.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.91e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.62e6iT - 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.41521056866922352739699692812, −10.02626372058317045094381046697, −9.785230604316507148269880127716, −8.147875998724711334885316466745, −6.74247875977839601909726477822, −5.52606068817150247430097149197, −4.10034323881524123053403134616, −3.20311795956196417359814856595, −1.64047053884867331293629281822, −0.01633767783950589532477397404,
1.56797347903765155504271958203, 2.96064725524879273282921873821, 5.04974416642390457340851883560, 6.05504070477773657328551698102, 6.94875580681397411570678612538, 7.81534251735269673193450205667, 8.690042421718821883168677271981, 10.48940191724471675878586194721, 11.42659399092357434302921462613, 12.20025987287249462373501826459
