L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 11-s + 13-s + 15-s − 6·19-s − 3·21-s − 5·23-s − 4·25-s − 27-s − 7·29-s − 33-s − 3·35-s + 2·37-s − 39-s + 3·41-s + 43-s − 45-s + 8·47-s + 2·49-s − 6·53-s − 55-s + 6·57-s − 3·59-s − 61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s − 1.37·19-s − 0.654·21-s − 1.04·23-s − 4/5·25-s − 0.192·27-s − 1.29·29-s − 0.174·33-s − 0.507·35-s + 0.328·37-s − 0.160·39-s + 0.468·41-s + 0.152·43-s − 0.149·45-s + 1.16·47-s + 2/7·49-s − 0.824·53-s − 0.134·55-s + 0.794·57-s − 0.390·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.010248762013426871554477468793, −7.70663629971428289917229765564, −6.65657138495982538759683022615, −5.94874166347674677810205177239, −5.20905896529497922014689386360, −4.21822889616307182698301301846, −3.90133019896858765147545146641, −2.31110490904312665239824967422, −1.45490216740932309441486631023, 0,
1.45490216740932309441486631023, 2.31110490904312665239824967422, 3.90133019896858765147545146641, 4.21822889616307182698301301846, 5.20905896529497922014689386360, 5.94874166347674677810205177239, 6.65657138495982538759683022615, 7.70663629971428289917229765564, 8.010248762013426871554477468793