| L(s) = 1 | − 82·5-s − 243·9-s − 1.19e3·13-s + 2.24e3·17-s + 3.59e3·25-s + 2.95e3·29-s − 1.22e4·37-s − 2.09e4·41-s + 1.99e4·45-s − 1.68e4·49-s + 7.29e3·53-s + 1.89e4·61-s + 9.79e4·65-s − 8.88e4·73-s + 5.90e4·81-s − 1.83e5·85-s + 5.10e4·89-s − 9.21e4·97-s − 9.80e4·101-s + 2.46e5·109-s + 1.18e5·113-s + 2.90e5·117-s + ⋯ |
| L(s) = 1 | − 1.46·5-s − 9-s − 1.95·13-s + 1.88·17-s + 1.15·25-s + 0.651·29-s − 1.47·37-s − 1.94·41-s + 1.46·45-s − 49-s + 0.356·53-s + 0.652·61-s + 2.87·65-s − 1.95·73-s + 81-s − 2.75·85-s + 0.683·89-s − 0.994·97-s − 0.955·101-s + 1.98·109-s + 0.874·113-s + 1.95·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + 82 T + p^{5} T^{2} \) |
| 7 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 + 1194 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2242 T + p^{5} T^{2} \) |
| 19 | \( 1 + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 - 2950 T + p^{5} T^{2} \) |
| 31 | \( 1 + p^{5} T^{2} \) |
| 37 | \( 1 + 12242 T + p^{5} T^{2} \) |
| 41 | \( 1 + 20950 T + p^{5} T^{2} \) |
| 43 | \( 1 + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 - 7294 T + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 - 18950 T + p^{5} T^{2} \) |
| 67 | \( 1 + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 + 88806 T + p^{5} T^{2} \) |
| 79 | \( 1 + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 - 51050 T + p^{5} T^{2} \) |
| 97 | \( 1 + 92142 T + p^{5} T^{2} \) |
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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
−15.07372016219682576556130554989, −14.28968853071208980500582768514, −12.24560925414745027315998309880, −11.75914005111003913086370594760, −10.09742600319534030534665317676, −8.334463836599475406821494026384, −7.29521090020210304006411105707, −5.09592723985540245460070195846, −3.22932183668501395921298198234, 0,
3.22932183668501395921298198234, 5.09592723985540245460070195846, 7.29521090020210304006411105707, 8.334463836599475406821494026384, 10.09742600319534030534665317676, 11.75914005111003913086370594760, 12.24560925414745027315998309880, 14.28968853071208980500582768514, 15.07372016219682576556130554989
