L(s) = 1 | − 5.42·2-s + 9·3-s − 2.52·4-s − 48.8·6-s + 29.0·7-s + 187.·8-s + 81·9-s + 121·11-s − 22.7·12-s + 433.·13-s − 157.·14-s − 936.·16-s − 681.·17-s − 439.·18-s + 1.40e3·19-s + 261.·21-s − 656.·22-s − 4.76e3·23-s + 1.68e3·24-s − 2.35e3·26-s + 729·27-s − 73.5·28-s − 646.·29-s + 840.·31-s − 913.·32-s + 1.08e3·33-s + 3.70e3·34-s + ⋯ |
L(s) = 1 | − 0.959·2-s + 0.577·3-s − 0.0789·4-s − 0.554·6-s + 0.224·7-s + 1.03·8-s + 0.333·9-s + 0.301·11-s − 0.0456·12-s + 0.710·13-s − 0.215·14-s − 0.914·16-s − 0.572·17-s − 0.319·18-s + 0.889·19-s + 0.129·21-s − 0.289·22-s − 1.87·23-s + 0.597·24-s − 0.682·26-s + 0.192·27-s − 0.0177·28-s − 0.142·29-s + 0.157·31-s − 0.157·32-s + 0.174·33-s + 0.548·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 5.42T + 32T^{2} \) |
| 7 | \( 1 - 29.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 433.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 681.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.40e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 646.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 840.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.57e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 857.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.88e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.28e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 790.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.72e4T + 8.58e9T^{2} \) |
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
−8.869525569500151738575670003565, −8.444217445183906023186651998469, −7.62956328317985924155909194383, −6.77442172325850792814211204743, −5.54743888807227237948353510731, −4.37015481105268111985262997330, −3.56601046268762620698431812613, −2.08936981584469315437627378977, −1.23480981603752261937894463893, 0,
1.23480981603752261937894463893, 2.08936981584469315437627378977, 3.56601046268762620698431812613, 4.37015481105268111985262997330, 5.54743888807227237948353510731, 6.77442172325850792814211204743, 7.62956328317985924155909194383, 8.444217445183906023186651998469, 8.869525569500151738575670003565