| L(s) = 1 | + 13·2-s + 41·4-s + 125·5-s + 1.38e3·7-s − 1.13e3·8-s + 1.62e3·10-s + 3.30e3·11-s + 8.50e3·13-s + 1.79e4·14-s − 1.99e4·16-s + 9.99e3·17-s + 4.12e4·19-s + 5.12e3·20-s + 4.29e4·22-s − 8.41e4·23-s + 1.56e4·25-s + 1.10e5·26-s + 5.65e4·28-s − 1.32e5·29-s − 5.58e4·31-s − 1.14e5·32-s + 1.29e5·34-s + 1.72e5·35-s + 2.28e5·37-s + 5.36e5·38-s − 1.41e5·40-s + 1.39e5·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.320·4-s + 0.447·5-s + 1.52·7-s − 0.780·8-s + 0.513·10-s + 0.748·11-s + 1.07·13-s + 1.74·14-s − 1.21·16-s + 0.493·17-s + 1.37·19-s + 0.143·20-s + 0.860·22-s − 1.44·23-s + 1/5·25-s + 1.23·26-s + 0.487·28-s − 1.01·29-s − 0.336·31-s − 0.618·32-s + 0.566·34-s + 0.680·35-s + 0.740·37-s + 1.58·38-s − 0.349·40-s + 0.316·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.780586143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.780586143\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 - 13 T + p^{7} T^{2} \) |
| 7 | \( 1 - 1380 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3304 T + p^{7} T^{2} \) |
| 13 | \( 1 - 8506 T + p^{7} T^{2} \) |
| 17 | \( 1 - 9994 T + p^{7} T^{2} \) |
| 19 | \( 1 - 41236 T + p^{7} T^{2} \) |
| 23 | \( 1 + 84120 T + p^{7} T^{2} \) |
| 29 | \( 1 + 132802 T + p^{7} T^{2} \) |
| 31 | \( 1 + 1800 p T + p^{7} T^{2} \) |
| 37 | \( 1 - 228170 T + p^{7} T^{2} \) |
| 41 | \( 1 - 139670 T + p^{7} T^{2} \) |
| 43 | \( 1 + 755492 T + p^{7} T^{2} \) |
| 47 | \( 1 + 836984 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1641650 T + p^{7} T^{2} \) |
| 59 | \( 1 - 989656 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1658162 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4523844 T + p^{7} T^{2} \) |
| 71 | \( 1 - 389408 T + p^{7} T^{2} \) |
| 73 | \( 1 - 5617330 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3901080 T + p^{7} T^{2} \) |
| 83 | \( 1 - 9394116 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2803746 T + p^{7} T^{2} \) |
| 97 | \( 1 - 5099426 T + p^{7} T^{2} \) |
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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
−14.19185227905231660164584811602, −13.49636910059816556979354209011, −12.02623962779372690441936402432, −11.22135112734841066600128877494, −9.420225573849127920782032468658, −8.009321999753543626583346784899, −6.10206747534084543211560833008, −5.00532291989611888146216748771, −3.64972560498620197937273757011, −1.56400471361453700371053305560,
1.56400471361453700371053305560, 3.64972560498620197937273757011, 5.00532291989611888146216748771, 6.10206747534084543211560833008, 8.009321999753543626583346784899, 9.420225573849127920782032468658, 11.22135112734841066600128877494, 12.02623962779372690441936402432, 13.49636910059816556979354209011, 14.19185227905231660164584811602
