| L(s) = 1 | + 60·4-s + 2.07e3·9-s + 344·11-s − 1.27e4·16-s + 5.18e4·19-s + 1.63e5·29-s − 3.13e5·31-s + 1.24e5·36-s + 9.35e5·41-s + 2.06e4·44-s − 1.05e6·49-s + 2.67e6·59-s − 1.84e6·61-s − 1.75e6·64-s + 1.02e7·71-s + 3.11e6·76-s + 1.92e3·79-s − 4.98e5·81-s − 4.02e6·89-s + 7.12e5·99-s + 1.94e7·101-s + 5.36e7·109-s + 9.79e6·116-s − 3.88e7·121-s − 1.88e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 0.468·4-s + 0.946·9-s + 0.0779·11-s − 0.780·16-s + 1.73·19-s + 1.24·29-s − 1.89·31-s + 0.443·36-s + 2.12·41-s + 0.0365·44-s − 1.28·49-s + 1.69·59-s − 1.04·61-s − 0.834·64-s + 3.38·71-s + 0.813·76-s + 0.000438·79-s − 0.104·81-s − 0.604·89-s + 0.0737·99-s + 1.87·101-s + 3.96·109-s + 0.582·116-s − 1.99·121-s − 0.886·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.692653282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.692653282\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 p^{2} T^{2} + p^{14} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 230 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 1055650 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 172 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 110581990 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 670516830 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 25940 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12554590 p^{2} T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 81610 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 156888 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 177736018390 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 467882 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 294428594950 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 855729331470 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 709747151670 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1337420 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 923978 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 11485729542230 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5103392 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3883428557710 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 960 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 16562284327030 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2010570 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 137763289375870 T^{2} + p^{14} T^{4} \) |
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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
−16.18478196156870312828106492811, −15.77586401400586956018855106681, −15.29634962832661054866023615961, −14.16082766145288064501529506762, −14.09452987362474525913466101564, −12.89951475999537219051234035554, −12.67495339055435890010227414413, −11.63469453383743054186628987156, −11.24391586259260561186100697700, −10.39193094788074765641838228124, −9.611466805722780882395064240993, −9.083942751012098043938708343412, −7.88819761440178297211863641977, −7.25360673551294845101201508074, −6.57330964574480363503174895963, −5.47822036430367079618258835068, −4.50176754307469675040540383347, −3.37731055173171707997652935919, −2.11462606779318106149353526097, −0.908484910533370124177627263998,
0.908484910533370124177627263998, 2.11462606779318106149353526097, 3.37731055173171707997652935919, 4.50176754307469675040540383347, 5.47822036430367079618258835068, 6.57330964574480363503174895963, 7.25360673551294845101201508074, 7.88819761440178297211863641977, 9.083942751012098043938708343412, 9.611466805722780882395064240993, 10.39193094788074765641838228124, 11.24391586259260561186100697700, 11.63469453383743054186628987156, 12.67495339055435890010227414413, 12.89951475999537219051234035554, 14.09452987362474525913466101564, 14.16082766145288064501529506762, 15.29634962832661054866023615961, 15.77586401400586956018855106681, 16.18478196156870312828106492811
