| L(s) = 1 | − 3-s + 9-s + 3·11-s + 3·13-s − 6·17-s − 5·19-s + 23-s − 27-s − 4·29-s + 10·31-s − 3·33-s + 37-s − 3·39-s + 3·41-s + 6·43-s + 47-s + 6·51-s − 7·53-s + 5·57-s − 4·59-s + 10·61-s + 2·67-s − 69-s + 2·71-s − 10·79-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.904·11-s + 0.832·13-s − 1.45·17-s − 1.14·19-s + 0.208·23-s − 0.192·27-s − 0.742·29-s + 1.79·31-s − 0.522·33-s + 0.164·37-s − 0.480·39-s + 0.468·41-s + 0.914·43-s + 0.145·47-s + 0.840·51-s − 0.961·53-s + 0.662·57-s − 0.520·59-s + 1.28·61-s + 0.244·67-s − 0.120·69-s + 0.237·71-s − 1.12·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.78310084727758, −13.25943881367163, −12.87393883322535, −12.41644777336801, −11.77992550300233, −11.38476614932688, −10.91049356844413, −10.68367687600242, −9.899495727620729, −9.453742046665812, −8.856962945564676, −8.535105185926275, −7.962806753874156, −7.223426853919295, −6.702784718599760, −6.247030840933898, −6.056847758986346, −5.201033478864722, −4.566960119601067, −4.142174722696990, −3.761989361422651, −2.806667380647617, −2.229685429859475, −1.498613866408758, −0.8565246556909590, 0,
0.8565246556909590, 1.498613866408758, 2.229685429859475, 2.806667380647617, 3.761989361422651, 4.142174722696990, 4.566960119601067, 5.201033478864722, 6.056847758986346, 6.247030840933898, 6.702784718599760, 7.223426853919295, 7.962806753874156, 8.535105185926275, 8.856962945564676, 9.453742046665812, 9.899495727620729, 10.68367687600242, 10.91049356844413, 11.38476614932688, 11.77992550300233, 12.41644777336801, 12.87393883322535, 13.25943881367163, 13.78310084727758
