| L(s) = 1 | + 6.22e4·3-s − 8.35e6·5-s + 1.26e9·7-s − 6.58e9·9-s + 6.86e10·11-s − 6.58e11·13-s − 5.20e11·15-s + 1.61e13·17-s − 3.29e13·19-s + 7.85e13·21-s − 1.17e14·23-s − 4.06e14·25-s − 1.06e15·27-s + 6.60e14·29-s − 4.65e15·31-s + 4.27e15·33-s − 1.05e16·35-s + 2.72e16·37-s − 4.09e16·39-s − 1.34e17·41-s − 3.14e16·43-s + 5.50e16·45-s + 4.70e17·47-s + 1.03e18·49-s + 1.00e18·51-s + 2.07e17·53-s − 5.73e17·55-s + ⋯ |
| L(s) = 1 | + 0.608·3-s − 0.382·5-s + 1.69·7-s − 0.629·9-s + 0.798·11-s − 1.32·13-s − 0.232·15-s + 1.94·17-s − 1.23·19-s + 1.02·21-s − 0.593·23-s − 0.853·25-s − 0.991·27-s + 0.291·29-s − 1.01·31-s + 0.485·33-s − 0.647·35-s + 0.932·37-s − 0.805·39-s − 1.56·41-s − 0.222·43-s + 0.241·45-s + 1.30·47-s + 1.85·49-s + 1.18·51-s + 0.162·53-s − 0.305·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 6.22e4T + 1.04e10T^{2} \) |
| 5 | \( 1 + 8.35e6T + 4.76e14T^{2} \) |
| 7 | \( 1 - 1.26e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 6.86e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.58e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.61e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.29e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.17e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 6.60e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 4.65e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.72e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.34e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 3.14e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.70e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.07e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 4.95e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 3.82e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.61e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.13e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.99e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 6.63e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 4.03e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.05e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 3.31e20T + 5.27e41T^{2} \) |
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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
−10.38938410311021790868398176254, −9.105600049401408177269803071332, −8.044665896920995604385925554016, −7.55369232936741653646329701788, −5.78970297788477391196448074873, −4.69536347108401663946393124459, −3.63852122733545198377140639404, −2.31605574438379021797303163268, −1.43263554569570772938843389458, 0,
1.43263554569570772938843389458, 2.31605574438379021797303163268, 3.63852122733545198377140639404, 4.69536347108401663946393124459, 5.78970297788477391196448074873, 7.55369232936741653646329701788, 8.044665896920995604385925554016, 9.105600049401408177269803071332, 10.38938410311021790868398176254
