L(s) = 1 | − 33.9i·5-s − 80.4i·7-s + 305.·11-s + 443.·13-s + 380. i·17-s + 434. i·19-s + 1.19e3·23-s + 1.97e3·25-s + 1.44e3i·29-s + 4.92e3i·31-s − 2.73e3·35-s − 2.08e3·37-s − 1.44e4i·41-s − 6.00e3i·43-s + 7.55e3·47-s + ⋯ |
L(s) = 1 | − 0.607i·5-s − 0.620i·7-s + 0.761·11-s + 0.727·13-s + 0.319i·17-s + 0.276i·19-s + 0.471·23-s + 0.631·25-s + 0.318i·29-s + 0.921i·31-s − 0.377·35-s − 0.250·37-s − 1.34i·41-s − 0.495i·43-s + 0.499·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.318337514\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.318337514\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 33.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 80.4iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 305.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 443.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 380. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 434. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.44e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.92e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.44e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 6.00e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.55e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.32e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 6.22e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.38e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.89e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.19e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.27e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 2.93e4T + 8.58e9T^{2} \) |
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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
−10.31742266411019574603952194677, −9.079090332380000724640491698327, −8.589456972379940581518316244834, −7.37276292139334923160335414393, −6.49944458140506263195112855640, −5.37003187722905809445182247558, −4.27952109765306340152846898997, −3.37371214354329411644792570253, −1.64138959802081862878444502853, −0.69476006831408268057893468683,
1.01935407474383196545684175934, 2.43249869015279287098140335968, 3.46479274836473408383354234873, 4.68324676506636577942319024689, 5.95036481097697898449964841381, 6.65247139197725272503130456862, 7.71667141326601650999449027480, 8.819692155302317418177310918618, 9.468883529129541991549088821085, 10.61992387014696765489337201140