L(s) = 1 | − 2-s + 4-s − 2·5-s − 2·7-s − 8-s + 2·10-s + 11-s + 4·13-s + 2·14-s + 16-s − 17-s + 2·19-s − 2·20-s − 22-s − 2·23-s − 25-s − 4·26-s − 2·28-s − 2·29-s − 4·31-s − 32-s + 34-s + 4·35-s + 6·37-s − 2·38-s + 2·40-s − 6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s − 0.447·20-s − 0.213·22-s − 0.417·23-s − 1/5·25-s − 0.784·26-s − 0.377·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.676·35-s + 0.986·37-s − 0.324·38-s + 0.316·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−8.314263977249175248487166556648, −7.58325353722904988213090916409, −6.88058020825979319725915000308, −6.18875213971508029398489255414, −5.36978935157106081419405981004, −3.96857785975134090094645009685, −3.63966168110760500189322356889, −2.50328229742078417917317816036, −1.21053447099632842174411855421, 0,
1.21053447099632842174411855421, 2.50328229742078417917317816036, 3.63966168110760500189322356889, 3.96857785975134090094645009685, 5.36978935157106081419405981004, 6.18875213971508029398489255414, 6.88058020825979319725915000308, 7.58325353722904988213090916409, 8.314263977249175248487166556648