L(s) = 1 | + 5·4-s + 8·11-s + 12·16-s − 20·29-s + 12·31-s − 28·41-s + 40·44-s + 20·59-s + 12·61-s + 15·64-s + 8·71-s + 2·81-s + 60·89-s − 28·101-s + 20·109-s − 100·116-s + 6·121-s + 60·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 140·164-s + ⋯ |
L(s) = 1 | + 5/2·4-s + 2.41·11-s + 3·16-s − 3.71·29-s + 2.15·31-s − 4.37·41-s + 6.03·44-s + 2.60·59-s + 1.53·61-s + 15/8·64-s + 0.949·71-s + 2/9·81-s + 6.35·89-s − 2.78·101-s + 1.91·109-s − 9.28·116-s + 6/11·121-s + 5.38·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 10.9·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.333920186\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.333920186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1363 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 7603 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 250 T^{2} + 24523 T^{4} - 250 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 8638 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 27998 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.77910585402552743724069145297, −6.68024070795720090866326855678, −6.66978319234314559057034129782, −6.39918621526094427027118769606, −6.28322838270318941266615162931, −6.01913710442662557314706496631, −5.67604785482192682849773629809, −5.33982425277558915744616425653, −5.23301211118028482926721848131, −5.11337528885542241145017963819, −4.80024566876612820989611811789, −4.21520229611882138490212393104, −4.08088010697823706775331034368, −3.90958486544072445413140700674, −3.58143781192695999451475822173, −3.39551855399943895349341642124, −3.26315895062392405127863435071, −2.89428874118669310415636588789, −2.40386594359924373405291608600, −2.20391730363459160834634347654, −1.91100396554140594942435489984, −1.76600401765979428066316529703, −1.56446259368948466073359165452, −0.981588788190884553490601443660, −0.58018497838635622608429375804,
0.58018497838635622608429375804, 0.981588788190884553490601443660, 1.56446259368948466073359165452, 1.76600401765979428066316529703, 1.91100396554140594942435489984, 2.20391730363459160834634347654, 2.40386594359924373405291608600, 2.89428874118669310415636588789, 3.26315895062392405127863435071, 3.39551855399943895349341642124, 3.58143781192695999451475822173, 3.90958486544072445413140700674, 4.08088010697823706775331034368, 4.21520229611882138490212393104, 4.80024566876612820989611811789, 5.11337528885542241145017963819, 5.23301211118028482926721848131, 5.33982425277558915744616425653, 5.67604785482192682849773629809, 6.01913710442662557314706496631, 6.28322838270318941266615162931, 6.39918621526094427027118769606, 6.66978319234314559057034129782, 6.68024070795720090866326855678, 6.77910585402552743724069145297