| L(s) = 1 | − 9·3-s − 21.1·5-s − 241.·7-s + 81·9-s + 641.·11-s + 316.·13-s + 190.·15-s − 901.·17-s − 2.16e3·19-s + 2.17e3·21-s + 397.·23-s − 2.67e3·25-s − 729·27-s + 5.17e3·29-s + 2.39e3·31-s − 5.77e3·33-s + 5.09e3·35-s + 1.35e4·37-s − 2.84e3·39-s + 1.10e4·41-s − 1.69e4·43-s − 1.71e3·45-s + 2.26e4·47-s + 4.13e4·49-s + 8.11e3·51-s + 7.71e3·53-s − 1.35e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·5-s − 1.86·7-s + 0.333·9-s + 1.59·11-s + 0.519·13-s + 0.218·15-s − 0.756·17-s − 1.37·19-s + 1.07·21-s + 0.156·23-s − 0.857·25-s − 0.192·27-s + 1.14·29-s + 0.446·31-s − 0.923·33-s + 0.702·35-s + 1.62·37-s − 0.299·39-s + 1.02·41-s − 1.39·43-s − 0.125·45-s + 1.49·47-s + 2.46·49-s + 0.436·51-s + 0.377·53-s − 0.604·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| good | 5 | \( 1 + 21.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 241.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 641.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 316.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 901.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 397.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.69e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.71e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.59e4T + 8.58e9T^{2} \) |
| 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
−9.240368111414741184298332195684, −8.456710701411553918422417565420, −7.03534615399561064634260735961, −6.40538401328410834925881669638, −6.01601873762511092654435524312, −4.28329303678356244197492857666, −3.83641900805572710623015173597, −2.56310353243379249460569200159, −0.991974157558748293335408950532, 0,
0.991974157558748293335408950532, 2.56310353243379249460569200159, 3.83641900805572710623015173597, 4.28329303678356244197492857666, 6.01601873762511092654435524312, 6.40538401328410834925881669638, 7.03534615399561064634260735961, 8.456710701411553918422417565420, 9.240368111414741184298332195684
