| L(s) = 1 | − 2-s − 4-s + 2·5-s − 4·7-s + 3·8-s − 3·9-s − 2·10-s − 2·13-s + 4·14-s − 16-s + 3·18-s − 4·19-s − 2·20-s − 4·23-s − 25-s + 2·26-s + 4·28-s − 6·29-s − 4·31-s − 5·32-s − 8·35-s + 3·36-s + 2·37-s + 4·38-s + 6·40-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.51·7-s + 1.06·8-s − 9-s − 0.632·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.707·18-s − 0.917·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.883·32-s − 1.35·35-s + 1/2·36-s + 0.328·37-s + 0.648·38-s + 0.948·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−11.02941934245556844339132034347, −10.04009807089741531902747917178, −9.478907532433233918366672610314, −8.796016188301955158560715963145, −7.59089249699874684434940494125, −6.30307949005461764625435345620, −5.50273032332247760535071006049, −3.87810234950044466939896282289, −2.34450910132149137086379940903, 0,
2.34450910132149137086379940903, 3.87810234950044466939896282289, 5.50273032332247760535071006049, 6.30307949005461764625435345620, 7.59089249699874684434940494125, 8.796016188301955158560715963145, 9.478907532433233918366672610314, 10.04009807089741531902747917178, 11.02941934245556844339132034347
