| L(s) = 1 | + 8·2-s − 87·3-s + 64·4-s + 321·5-s − 696·6-s − 181·7-s + 512·8-s + 5.38e3·9-s + 2.56e3·10-s + 7.78e3·11-s − 5.56e3·12-s + 2.19e3·13-s − 1.44e3·14-s − 2.79e4·15-s + 4.09e3·16-s + 9.06e3·17-s + 4.30e4·18-s − 3.71e4·19-s + 2.05e4·20-s + 1.57e4·21-s + 6.22e4·22-s + 1.90e4·23-s − 4.45e4·24-s + 2.49e4·25-s + 1.75e4·26-s − 2.77e5·27-s − 1.15e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 1/2·4-s + 1.14·5-s − 1.31·6-s − 0.199·7-s + 0.353·8-s + 2.46·9-s + 0.812·10-s + 1.76·11-s − 0.930·12-s + 0.277·13-s − 0.141·14-s − 2.13·15-s + 1/4·16-s + 0.447·17-s + 1.74·18-s − 1.24·19-s + 0.574·20-s + 0.371·21-s + 1.24·22-s + 0.325·23-s − 0.657·24-s + 0.318·25-s + 0.196·26-s − 2.71·27-s − 0.0997·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.847202785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847202785\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 13 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 + 29 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 321 T + p^{7} T^{2} \) |
| 7 | \( 1 + 181 T + p^{7} T^{2} \) |
| 11 | \( 1 - 7782 T + p^{7} T^{2} \) |
| 17 | \( 1 - 9069 T + p^{7} T^{2} \) |
| 19 | \( 1 + 37150 T + p^{7} T^{2} \) |
| 23 | \( 1 - 19008 T + p^{7} T^{2} \) |
| 29 | \( 1 - 174750 T + p^{7} T^{2} \) |
| 31 | \( 1 - 29012 T + p^{7} T^{2} \) |
| 37 | \( 1 - 323669 T + p^{7} T^{2} \) |
| 41 | \( 1 - 795312 T + p^{7} T^{2} \) |
| 43 | \( 1 + 314137 T + p^{7} T^{2} \) |
| 47 | \( 1 + 447441 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1469232 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1627770 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2399608 T + p^{7} T^{2} \) |
| 67 | \( 1 + 64066 T + p^{7} T^{2} \) |
| 71 | \( 1 + 322383 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4454782 T + p^{7} T^{2} \) |
| 79 | \( 1 - 753560 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1219092 T + p^{7} T^{2} \) |
| 89 | \( 1 - 3390330 T + p^{7} T^{2} \) |
| 97 | \( 1 - 1628774 T + p^{7} T^{2} \) |
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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
−16.21283223838623658031062621141, −14.52606818858193387418708209882, −13.08108880815631645479012076645, −12.06288489059628952218744748825, −10.96247399378092781974030817554, −9.707502403325843454586535582075, −6.53559471469993015818510121343, −6.03489597522478265562738267179, −4.47123091425630715880737541735, −1.31960856305599856038970578627,
1.31960856305599856038970578627, 4.47123091425630715880737541735, 6.03489597522478265562738267179, 6.53559471469993015818510121343, 9.707502403325843454586535582075, 10.96247399378092781974030817554, 12.06288489059628952218744748825, 13.08108880815631645479012076645, 14.52606818858193387418708209882, 16.21283223838623658031062621141
