L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s + 7-s − 2·9-s − 2·10-s − 3·11-s − 2·12-s − 13-s − 2·14-s − 15-s − 4·16-s + 3·17-s + 4·18-s − 19-s + 2·20-s − 21-s + 6·22-s + 4·23-s + 25-s + 2·26-s + 5·27-s + 2·28-s + 5·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s − 2/3·9-s − 0.632·10-s − 0.904·11-s − 0.577·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s − 16-s + 0.727·17-s + 0.942·18-s − 0.229·19-s + 0.447·20-s − 0.218·21-s + 1.27·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.962·27-s + 0.377·28-s + 0.928·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−10.32221739075337598059175896889, −9.114687171684059628825134597073, −8.528252783263885056303576092770, −7.62859188111319130791942184895, −6.78846826777664281836508057923, −5.58069200719717366719140293254, −4.87135521086630417484723927522, −2.94785666765784560985425588519, −1.58957591968619034979500805363, 0,
1.58957591968619034979500805363, 2.94785666765784560985425588519, 4.87135521086630417484723927522, 5.58069200719717366719140293254, 6.78846826777664281836508057923, 7.62859188111319130791942184895, 8.528252783263885056303576092770, 9.114687171684059628825134597073, 10.32221739075337598059175896889