| L(s) = 1 | − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 4·11-s + 2·13-s + 14-s − 16-s + 6·17-s − 2·20-s + 4·22-s − 25-s − 2·26-s + 28-s − 2·29-s − 5·32-s − 6·34-s − 2·35-s − 6·37-s + 6·40-s + 2·41-s − 4·43-s + 4·44-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.986·37-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.603·44-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22743 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−16.03370510653655, −15.20086476300314, −14.70250827964801, −13.88078671543458, −13.65358953082019, −13.19018578387675, −12.61006758034389, −12.05574406630107, −11.18431424540378, −10.46703071482560, −10.26725378626327, −9.663185835132306, −9.294790128672518, −8.485664246907082, −8.111657677366945, −7.479347613770856, −6.863547300305273, −5.967107539671437, −5.422330053826607, −5.113520368931750, −4.056421545440046, −3.438774026611305, −2.589864309275685, −1.764511385695925, −1.001016967903345, 0,
1.001016967903345, 1.764511385695925, 2.589864309275685, 3.438774026611305, 4.056421545440046, 5.113520368931750, 5.422330053826607, 5.967107539671437, 6.863547300305273, 7.479347613770856, 8.111657677366945, 8.485664246907082, 9.294790128672518, 9.663185835132306, 10.26725378626327, 10.46703071482560, 11.18431424540378, 12.05574406630107, 12.61006758034389, 13.19018578387675, 13.65358953082019, 13.88078671543458, 14.70250827964801, 15.20086476300314, 16.03370510653655
