| L(s) = 1 | − 2·4-s + 2·5-s + 7-s − 4·13-s + 4·16-s − 17-s − 7·19-s − 4·20-s − 25-s − 2·28-s − 29-s − 3·31-s + 2·35-s + 8·37-s − 6·41-s − 12·43-s − 8·47-s + 49-s + 8·52-s + 6·53-s − 8·59-s − 13·61-s − 8·64-s − 8·65-s + 67-s + 2·68-s − 11·73-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s + 0.377·7-s − 1.10·13-s + 16-s − 0.242·17-s − 1.60·19-s − 0.894·20-s − 1/5·25-s − 0.377·28-s − 0.185·29-s − 0.538·31-s + 0.338·35-s + 1.31·37-s − 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 1.10·52-s + 0.824·53-s − 1.04·59-s − 1.66·61-s − 64-s − 0.992·65-s + 0.122·67-s + 0.242·68-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.40223571191151, −13.06897888589063, −12.66857775620220, −12.06134440479932, −11.69755764723322, −10.99726344588289, −10.49645462966850, −10.07791628626079, −9.726510120074051, −9.221894062371248, −8.820206037545403, −8.292303755486947, −7.874540399583287, −7.326532993176729, −6.647219653719628, −6.178804111785584, −5.722228274663432, −5.086898916834527, −4.715901063155706, −4.345533961858258, −3.650215560732966, −3.014190070607411, −2.306545177533922, −1.815920846156140, −1.267601101525550, 0, 0,
1.267601101525550, 1.815920846156140, 2.306545177533922, 3.014190070607411, 3.650215560732966, 4.345533961858258, 4.715901063155706, 5.086898916834527, 5.722228274663432, 6.178804111785584, 6.647219653719628, 7.326532993176729, 7.874540399583287, 8.292303755486947, 8.820206037545403, 9.221894062371248, 9.726510120074051, 10.07791628626079, 10.49645462966850, 10.99726344588289, 11.69755764723322, 12.06134440479932, 12.66857775620220, 13.06897888589063, 13.40223571191151
