| L(s) = 1 | − 2·5-s − 3·9-s + 2·11-s + 17-s − 2·19-s − 8·23-s − 25-s − 8·31-s + 4·37-s + 6·41-s − 4·43-s + 6·45-s − 8·47-s + 6·53-s − 4·55-s + 10·59-s + 10·61-s − 8·67-s + 4·71-s + 10·73-s − 4·79-s + 9·81-s − 6·83-s − 2·85-s + 6·89-s + 4·95-s + 14·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 0.603·11-s + 0.242·17-s − 0.458·19-s − 1.66·23-s − 1/5·25-s − 1.43·31-s + 0.657·37-s + 0.937·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s + 0.824·53-s − 0.539·55-s + 1.30·59-s + 1.28·61-s − 0.977·67-s + 0.474·71-s + 1.17·73-s − 0.450·79-s + 81-s − 0.658·83-s − 0.216·85-s + 0.635·89-s + 0.410·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.66559941031881, −14.32373250837898, −13.82366160707698, −13.05172172571249, −12.68696013858719, −11.95356054034812, −11.62156134499842, −11.33904658035005, −10.69903892675622, −9.992634830482615, −9.600277508236628, −8.731679060956944, −8.578003805151304, −7.797269696313333, −7.570759862108495, −6.737370724764039, −6.159135047641985, −5.702159888429872, −5.038299455131262, −4.244550258615855, −3.757253208071254, −3.371183077236614, −2.384355717048205, −1.881858896965633, −0.7390219862791393, 0,
0.7390219862791393, 1.881858896965633, 2.384355717048205, 3.371183077236614, 3.757253208071254, 4.244550258615855, 5.038299455131262, 5.702159888429872, 6.159135047641985, 6.737370724764039, 7.570759862108495, 7.797269696313333, 8.578003805151304, 8.731679060956944, 9.600277508236628, 9.992634830482615, 10.69903892675622, 11.33904658035005, 11.62156134499842, 11.95356054034812, 12.68696013858719, 13.05172172571249, 13.82366160707698, 14.32373250837898, 14.66559941031881
