L(s) = 1 | − 1.41·2-s + 0.414·3-s − 5-s − 0.585·6-s − 0.585·7-s + 2.82·8-s − 2.82·9-s + 1.41·10-s − 11-s − 6.24·13-s + 0.828·14-s − 0.414·15-s − 4.00·16-s + 0.585·17-s + 4.00·18-s − 19-s − 0.242·21-s + 1.41·22-s − 3·23-s + 1.17·24-s − 4·25-s + 8.82·26-s − 2.41·27-s + 2.24·29-s + 0.585·30-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 0.239·3-s − 0.447·5-s − 0.239·6-s − 0.221·7-s + 0.999·8-s − 0.942·9-s + 0.447·10-s − 0.301·11-s − 1.73·13-s + 0.221·14-s − 0.106·15-s − 1.00·16-s + 0.142·17-s + 0.942·18-s − 0.229·19-s − 0.0529·21-s + 0.301·22-s − 0.625·23-s + 0.239·24-s − 0.800·25-s + 1.73·26-s − 0.464·27-s + 0.416·29-s + 0.106·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 0.585T + 17T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 - 3.07T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 0.585T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 0.414T + 97T^{2} \) |
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
−11.78528494648351285918908043889, −10.67603568548032238096009848013, −9.760607320506683258382618585141, −8.961927499800109227389829265358, −7.942479983460655289164991279617, −7.29991824030977858611988412521, −5.58376600789641197137620065565, −4.22551461551423115358213443029, −2.48727755494252447957045829685, 0,
2.48727755494252447957045829685, 4.22551461551423115358213443029, 5.58376600789641197137620065565, 7.29991824030977858611988412521, 7.942479983460655289164991279617, 8.961927499800109227389829265358, 9.760607320506683258382618585141, 10.67603568548032238096009848013, 11.78528494648351285918908043889