| L(s) = 1 | − 9·3-s − 62.9·5-s − 85.2·7-s + 81·9-s + 69.5·11-s + 169·13-s + 566.·15-s − 1.37e3·17-s − 1.24e3·19-s + 767.·21-s − 749.·23-s + 843.·25-s − 729·27-s − 2.08e3·29-s − 3.86e3·31-s − 626.·33-s + 5.36e3·35-s − 1.47e4·37-s − 1.52e3·39-s − 1.03e4·41-s − 6.75e3·43-s − 5.10e3·45-s + 6.80e3·47-s − 9.54e3·49-s + 1.23e4·51-s − 2.99e4·53-s − 4.38e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.12·5-s − 0.657·7-s + 0.333·9-s + 0.173·11-s + 0.277·13-s + 0.650·15-s − 1.15·17-s − 0.788·19-s + 0.379·21-s − 0.295·23-s + 0.269·25-s − 0.192·27-s − 0.459·29-s − 0.722·31-s − 0.100·33-s + 0.740·35-s − 1.76·37-s − 0.160·39-s − 0.965·41-s − 0.557·43-s − 0.375·45-s + 0.449·47-s − 0.567·49-s + 0.665·51-s − 1.46·53-s − 0.195·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2829873667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2829873667\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 13 | \( 1 - 169T \) |
| good | 5 | \( 1 + 62.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 85.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 69.5T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.99e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.03e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.53e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e5T + 8.58e9T^{2} \) |
| 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
−9.924766961703708938216275619687, −8.890086742764119255999903227311, −8.099336690945009019867822512530, −6.98300354882576335442442866286, −6.44661797476780665264183582937, −5.21910054009892824908019614861, −4.14721621070103243218666131538, −3.43923980181687375625367081222, −1.87505753365457115495476573848, −0.25353645454652869299121311809,
0.25353645454652869299121311809, 1.87505753365457115495476573848, 3.43923980181687375625367081222, 4.14721621070103243218666131538, 5.21910054009892824908019614861, 6.44661797476780665264183582937, 6.98300354882576335442442866286, 8.099336690945009019867822512530, 8.890086742764119255999903227311, 9.924766961703708938216275619687
