| L(s) = 1 | − 3·3-s − 2·4-s − 5-s + 7-s + 6·9-s + 6·12-s + 4·13-s + 3·15-s + 4·16-s − 2·17-s + 6·19-s + 2·20-s − 3·21-s − 5·23-s − 4·25-s − 9·27-s − 2·28-s − 10·29-s + 31-s − 35-s − 12·36-s − 5·37-s − 12·39-s + 2·41-s + 8·43-s − 6·45-s + 8·47-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.73·12-s + 1.10·13-s + 0.774·15-s + 16-s − 0.485·17-s + 1.37·19-s + 0.447·20-s − 0.654·21-s − 1.04·23-s − 4/5·25-s − 1.73·27-s − 0.377·28-s − 1.85·29-s + 0.179·31-s − 0.169·35-s − 2·36-s − 0.821·37-s − 1.92·39-s + 0.312·41-s + 1.21·43-s − 0.894·45-s + 1.16·47-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 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
| 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
−9.901287739646014110658752080721, −9.057617126467774810255940907586, −7.956327480837723148155835866573, −7.17057178520211720164419084101, −5.80118601537640384820432803021, −5.58596676881303008604275522263, −4.39514744576497898862871693801, −3.79024025385094143595808125256, −1.32889476078570135837429482480, 0,
1.32889476078570135837429482480, 3.79024025385094143595808125256, 4.39514744576497898862871693801, 5.58596676881303008604275522263, 5.80118601537640384820432803021, 7.17057178520211720164419084101, 7.956327480837723148155835866573, 9.057617126467774810255940907586, 9.901287739646014110658752080721
