| L(s) = 1 | − 194.·3-s − 920.·5-s − 5.35e3·7-s + 1.80e4·9-s − 7.92e4·11-s + 2.85e4·13-s + 1.78e5·15-s + 4.52e5·17-s − 2.12e5·19-s + 1.04e6·21-s + 7.59e5·23-s − 1.10e6·25-s + 3.15e5·27-s − 9.00e5·29-s − 2.27e6·31-s + 1.54e7·33-s + 4.93e6·35-s − 4.70e6·37-s − 5.54e6·39-s + 3.39e7·41-s + 2.33e7·43-s − 1.66e7·45-s + 5.14e7·47-s − 1.16e7·49-s − 8.78e7·51-s + 1.01e8·53-s + 7.29e7·55-s + ⋯ |
| L(s) = 1 | − 1.38·3-s − 0.658·5-s − 0.843·7-s + 0.917·9-s − 1.63·11-s + 0.277·13-s + 0.911·15-s + 1.31·17-s − 0.374·19-s + 1.16·21-s + 0.565·23-s − 0.566·25-s + 0.114·27-s − 0.236·29-s − 0.441·31-s + 2.26·33-s + 0.555·35-s − 0.412·37-s − 0.384·39-s + 1.87·41-s + 1.04·43-s − 0.604·45-s + 1.53·47-s − 0.288·49-s − 1.81·51-s + 1.75·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 194.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 920.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.92e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.52e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.59e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 9.00e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.39e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.33e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.14e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.23e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.15e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.06e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.44e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.03e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.20e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.91e8T + 7.60e17T^{2} \) |
| 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
−10.60355406415164528432230989568, −9.516816150832771484575049687336, −8.011033037734526426908897214292, −7.16455700611799142717693917448, −5.90425844863114287575812582517, −5.33480214382167014845621576701, −3.99912029286932311295399467493, −2.72932067970497987323170898050, −0.816540532126358517243907746039, 0,
0.816540532126358517243907746039, 2.72932067970497987323170898050, 3.99912029286932311295399467493, 5.33480214382167014845621576701, 5.90425844863114287575812582517, 7.16455700611799142717693917448, 8.011033037734526426908897214292, 9.516816150832771484575049687336, 10.60355406415164528432230989568
