L(s) = 1 | − 504·11-s − 329·16-s − 2.60e3·31-s − 1.26e4·41-s − 2.31e4·61-s − 2.04e4·71-s + 2.28e4·101-s + 1.00e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 1.65e5·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | − 4.16·11-s − 1.28·16-s − 2.70·31-s − 7.49·41-s − 6.21·61-s − 4.05·71-s + 2.24·101-s + 6.84·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.34e−5·173-s + 5.35·176-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.002138185340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002138185340\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^3$ | \( 1 + 329 T^{4} + p^{16} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 7111198 T^{4} + p^{16} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 126 T + p^{4} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 1302619486 T^{4} + p^{16} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 133512382 T^{4} + p^{16} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 256798 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 151272063362 T^{4} + p^{16} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 265054 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 650 T + p^{4} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 1642456827358 T^{4} + p^{16} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 3150 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 22531122583102 T^{4} + p^{16} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 35229109920578 T^{4} + p^{16} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 90369826510078 T^{4} + p^{16} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 24195518 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5782 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 724831933536382 T^{4} + p^{16} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 72 p T + p^{4} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 16791040992962 T^{4} + p^{16} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 71799262 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 3708377573377154 T^{4} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 124356638 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 6777796734182978 T^{4} + p^{16} T^{8} \) |
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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
−8.236939576822532525904015451546, −8.001413353054781651855505821154, −7.72704110018025414752065254378, −7.39857102521711232765998062462, −7.28492797867781763682568460608, −6.91733046046314980654817511112, −6.84928218200705188487274217078, −6.29021040478947080666852137481, −5.79481542997291438200274128365, −5.78604939165953174994424769418, −5.52535323271009920402198717624, −4.97371417671952871869925077896, −4.80430472328005551342448989201, −4.76267430746406656164794591676, −4.62527454204075695807900430819, −3.65975757167091870198037519205, −3.43650138662076897022217161516, −3.15689903233654401698360164108, −2.88363102684553699960662801936, −2.55861670069233308514166072699, −1.93782618587624118618432793180, −1.67677521468806568350362145596, −1.59282506370957048088407604154, −0.07617712722087053367727273925, −0.06271255710838321324271480802,
0.06271255710838321324271480802, 0.07617712722087053367727273925, 1.59282506370957048088407604154, 1.67677521468806568350362145596, 1.93782618587624118618432793180, 2.55861670069233308514166072699, 2.88363102684553699960662801936, 3.15689903233654401698360164108, 3.43650138662076897022217161516, 3.65975757167091870198037519205, 4.62527454204075695807900430819, 4.76267430746406656164794591676, 4.80430472328005551342448989201, 4.97371417671952871869925077896, 5.52535323271009920402198717624, 5.78604939165953174994424769418, 5.79481542997291438200274128365, 6.29021040478947080666852137481, 6.84928218200705188487274217078, 6.91733046046314980654817511112, 7.28492797867781763682568460608, 7.39857102521711232765998062462, 7.72704110018025414752065254378, 8.001413353054781651855505821154, 8.236939576822532525904015451546