| L(s) = 1 | + 9·3-s − 20.4·5-s − 35.4·7-s + 81·9-s + 486.·11-s − 200.·13-s − 184.·15-s − 692.·17-s − 2.64e3·19-s − 318.·21-s + 3.62e3·23-s − 2.70e3·25-s + 729·27-s + 8.49e3·29-s + 6.26e3·31-s + 4.38e3·33-s + 725.·35-s − 9.59e3·37-s − 1.80e3·39-s + 3.77e3·41-s + 5.06e3·43-s − 1.65e3·45-s − 1.22e4·47-s − 1.55e4·49-s − 6.23e3·51-s + 1.68e4·53-s − 9.97e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.366·5-s − 0.273·7-s + 0.333·9-s + 1.21·11-s − 0.329·13-s − 0.211·15-s − 0.581·17-s − 1.67·19-s − 0.157·21-s + 1.43·23-s − 0.865·25-s + 0.192·27-s + 1.87·29-s + 1.17·31-s + 0.700·33-s + 0.100·35-s − 1.15·37-s − 0.190·39-s + 0.350·41-s + 0.417·43-s − 0.122·45-s − 0.807·47-s − 0.925·49-s − 0.335·51-s + 0.824·53-s − 0.444·55-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
| \(L(3)\) |
\(\approx\) |
\(2.465769012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465769012\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 20.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 35.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 200.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 692.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.59e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.77e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.22e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.87e4T + 8.58e9T^{2} \) |
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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
−9.419149988628101172546419680385, −8.706142587504366943875513751302, −8.037756975720185285192970846938, −6.75810954855759077052800820526, −6.45806456915900032698380615685, −4.81148328089318065063336978516, −4.08530992464791282404537975190, −3.06646941084177212729259863322, −1.98039010393660317139663110381, −0.70180910789587226150465748416,
0.70180910789587226150465748416, 1.98039010393660317139663110381, 3.06646941084177212729259863322, 4.08530992464791282404537975190, 4.81148328089318065063336978516, 6.45806456915900032698380615685, 6.75810954855759077052800820526, 8.037756975720185285192970846938, 8.706142587504366943875513751302, 9.419149988628101172546419680385
