L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·5-s − 2·6-s + 7-s + 3·8-s + 9-s + 2·10-s − 2·12-s − 4·13-s − 14-s − 4·15-s − 16-s − 4·17-s − 18-s + 2·20-s + 2·21-s − 4·23-s + 6·24-s − 25-s + 4·26-s − 4·27-s − 28-s + 6·29-s + 4·30-s + 10·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.577·12-s − 1.10·13-s − 0.267·14-s − 1.03·15-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.447·20-s + 0.436·21-s − 0.834·23-s + 1.22·24-s − 1/5·25-s + 0.784·26-s − 0.769·27-s − 0.188·28-s + 1.11·29-s + 0.730·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−9.742618950410792459548371471551, −8.572504027942197515704035369998, −8.339912824949481407935374648788, −7.67378178112872223010860918103, −6.70583207072298392541889211635, −4.94559945005359926667122742610, −4.28286203006294711588571475853, −3.17103530919462504679231815837, −1.91939450772296709675161614006, 0,
1.91939450772296709675161614006, 3.17103530919462504679231815837, 4.28286203006294711588571475853, 4.94559945005359926667122742610, 6.70583207072298392541889211635, 7.67378178112872223010860918103, 8.339912824949481407935374648788, 8.572504027942197515704035369998, 9.742618950410792459548371471551