| L(s) = 1 | − 2·5-s − 7-s − 4·11-s + 2·13-s − 17-s + 4·19-s + 8·23-s − 25-s + 6·29-s + 2·35-s + 2·37-s − 10·41-s − 4·43-s + 49-s + 6·53-s + 8·55-s + 4·59-s − 6·61-s − 4·65-s − 12·67-s − 8·71-s − 6·73-s + 4·77-s + 12·83-s + 2·85-s + 6·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.20·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 0.338·35-s + 0.328·37-s − 1.56·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s − 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.455·77-s + 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.53827030149986, −13.65290571192975, −13.36249700646221, −13.12447097450825, −12.28079597249420, −11.91620847846825, −11.47072161458533, −10.87150162338248, −10.38795008369887, −10.02054894709672, −9.206667605662427, −8.773471031622498, −8.235641104339930, −7.712597565011548, −7.247501259765273, −6.722101536746946, −6.096924368743713, −5.342364817874684, −4.956067354820176, −4.336528144938842, −3.579802914646130, −3.065810801754088, −2.664769111545962, −1.595640583413932, −0.7957246217533100, 0,
0.7957246217533100, 1.595640583413932, 2.664769111545962, 3.065810801754088, 3.579802914646130, 4.336528144938842, 4.956067354820176, 5.342364817874684, 6.096924368743713, 6.722101536746946, 7.247501259765273, 7.712597565011548, 8.235641104339930, 8.773471031622498, 9.206667605662427, 10.02054894709672, 10.38795008369887, 10.87150162338248, 11.47072161458533, 11.91620847846825, 12.28079597249420, 13.12447097450825, 13.36249700646221, 13.65290571192975, 14.53827030149986
