| L(s) = 1 | + 10·3-s − 84·5-s − 49·7-s − 143·9-s − 336·11-s − 584·13-s − 840·15-s − 1.45e3·17-s + 470·19-s − 490·21-s + 4.20e3·23-s + 3.93e3·25-s − 3.86e3·27-s − 4.86e3·29-s + 7.37e3·31-s − 3.36e3·33-s + 4.11e3·35-s − 1.43e4·37-s − 5.84e3·39-s + 6.22e3·41-s + 3.70e3·43-s + 1.20e4·45-s + 1.81e3·47-s + 2.40e3·49-s − 1.45e4·51-s + 3.72e4·53-s + 2.82e4·55-s + ⋯ |
| L(s) = 1 | + 0.641·3-s − 1.50·5-s − 0.377·7-s − 0.588·9-s − 0.837·11-s − 0.958·13-s − 0.963·15-s − 1.22·17-s + 0.298·19-s − 0.242·21-s + 1.65·23-s + 1.25·25-s − 1.01·27-s − 1.07·29-s + 1.37·31-s − 0.537·33-s + 0.567·35-s − 1.72·37-s − 0.614·39-s + 0.578·41-s + 0.305·43-s + 0.884·45-s + 0.119·47-s + 1/7·49-s − 0.784·51-s + 1.82·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7610647875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7610647875\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 10 T + p^{5} T^{2} \) |
| 5 | \( 1 + 84 T + p^{5} T^{2} \) |
| 11 | \( 1 + 336 T + p^{5} T^{2} \) |
| 13 | \( 1 + 584 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 - 470 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 + 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 - 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 + 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 + 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 - 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 + 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 - 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 + 16978 T + p^{5} T^{2} \) |
| 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
−10.36967163749159040605141990520, −9.103771486776076927819529686218, −8.509303455395327914975411118864, −7.55138279047458077111305986847, −6.97271262994566968361497308331, −5.37304952811771359679510442128, −4.34539886878522758049469210761, −3.24632916437080182068903729735, −2.47853065889974630512971220113, −0.40807910288017114760233111705,
0.40807910288017114760233111705, 2.47853065889974630512971220113, 3.24632916437080182068903729735, 4.34539886878522758049469210761, 5.37304952811771359679510442128, 6.97271262994566968361497308331, 7.55138279047458077111305986847, 8.509303455395327914975411118864, 9.103771486776076927819529686218, 10.36967163749159040605141990520
