| L(s) = 1 | + 288.·3-s − 6.16e3·5-s − 1.68e4·7-s − 9.40e4·9-s + 1.33e5·11-s + 1.72e6·13-s − 1.77e6·15-s + 1.90e6·17-s + 1.35e7·19-s − 4.84e6·21-s + 3.53e7·23-s − 1.07e7·25-s − 7.81e7·27-s − 8.11e7·29-s + 8.72e7·31-s + 3.86e7·33-s + 1.03e8·35-s − 5.94e8·37-s + 4.98e8·39-s − 1.60e8·41-s − 1.44e9·43-s + 5.80e8·45-s − 2.49e9·47-s + 2.82e8·49-s + 5.47e8·51-s − 5.86e9·53-s − 8.26e8·55-s + ⋯ |
| L(s) = 1 | + 0.684·3-s − 0.882·5-s − 0.377·7-s − 0.530·9-s + 0.250·11-s + 1.29·13-s − 0.604·15-s + 0.324·17-s + 1.25·19-s − 0.258·21-s + 1.14·23-s − 0.220·25-s − 1.04·27-s − 0.734·29-s + 0.547·31-s + 0.171·33-s + 0.333·35-s − 1.40·37-s + 0.884·39-s − 0.216·41-s − 1.50·43-s + 0.468·45-s − 1.58·47-s + 0.142·49-s + 0.222·51-s − 1.92·53-s − 0.221·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 3 | \( 1 - 288.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 6.16e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 1.33e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.72e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.35e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.53e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 8.11e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 8.72e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.94e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.60e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.44e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.49e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.86e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.50e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.08e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.67e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 8.61e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.88e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.42e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.68e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.34e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.43e10T + 7.15e21T^{2} \) |
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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
−11.15572182075186864905742978015, −9.693508411129831883316677051848, −8.670480833119626877407167672143, −7.898492270921304261766085989946, −6.68464924700331389923956088851, −5.29935098892522133448319465597, −3.65085684519840372367600641907, −3.15351710763104598242358011335, −1.40061643488169871752168687403, 0,
1.40061643488169871752168687403, 3.15351710763104598242358011335, 3.65085684519840372367600641907, 5.29935098892522133448319465597, 6.68464924700331389923956088851, 7.898492270921304261766085989946, 8.670480833119626877407167672143, 9.693508411129831883316677051848, 11.15572182075186864905742978015
