| L(s) = 1 | − 1.45e3·5-s + 7.80e4·13-s + 1.31e5·17-s + 7.99e5·25-s − 4.04e5·29-s − 3.75e6·37-s − 6.18e6·41-s + 2.19e6·49-s + 2.13e6·53-s + 3.43e7·61-s − 1.13e8·65-s − 1.06e8·73-s − 1.91e8·85-s − 1.73e8·89-s − 1.47e8·97-s − 3.82e8·101-s + 1.37e8·109-s − 6.61e7·113-s + 2.52e8·121-s + 1.70e8·125-s + 127-s + 131-s + 137-s + 139-s + 5.86e8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.32·5-s + 2.73·13-s + 1.57·17-s + 2.04·25-s − 0.571·29-s − 2.00·37-s − 2.18·41-s + 0.380·49-s + 0.270·53-s + 2.47·61-s − 6.35·65-s − 3.75·73-s − 3.66·85-s − 2.76·89-s − 1.66·97-s − 3.67·101-s + 0.972·109-s − 0.405·113-s + 1.17·121-s + 0.700·125-s + 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5148691633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5148691633\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 726 T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 914 p^{4} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 252323090 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 39034 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 65814 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 17000201234 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 95455358590 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 202062 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 276239804882 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 1876030 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3091050 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 18251245763090 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 7214640194114 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 1066482 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 260442349515410 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 17154194 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59866697031314 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 295210326091390 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 53286014 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2700438986177234 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4443915113493650 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 86667234 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 73901822 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.24332434150722792388088570233, −11.23588028393465378454250933998, −11.21474285187317208649225378273, −10.22760799966659563517862059299, −10.07991383198158764479110911142, −8.785302677587838440953240196590, −8.681876505504639995946102334657, −8.131624285403376019852162878292, −7.81770812513077481574615372421, −6.99894372701087408169432808351, −6.76856177848605040873031603738, −5.55663056819985722961458157232, −5.54566028193437022397373300271, −4.23190489262732324403069674012, −3.91848734507428364204258636077, −3.47396882675293710578096509076, −3.07669278746858495999326622779, −1.52895582506519606598232654234, −1.18703710312566985900498957512, −0.20706452938897119757122387540,
0.20706452938897119757122387540, 1.18703710312566985900498957512, 1.52895582506519606598232654234, 3.07669278746858495999326622779, 3.47396882675293710578096509076, 3.91848734507428364204258636077, 4.23190489262732324403069674012, 5.54566028193437022397373300271, 5.55663056819985722961458157232, 6.76856177848605040873031603738, 6.99894372701087408169432808351, 7.81770812513077481574615372421, 8.131624285403376019852162878292, 8.681876505504639995946102334657, 8.785302677587838440953240196590, 10.07991383198158764479110911142, 10.22760799966659563517862059299, 11.21474285187317208649225378273, 11.23588028393465378454250933998, 12.24332434150722792388088570233
