| L(s) = 1 | − 4·5-s + 3·9-s − 6·11-s + 16·19-s + 5·25-s + 4·29-s − 4·31-s + 24·41-s − 12·45-s + 24·55-s + 20·59-s − 14·79-s + 9·81-s − 16·89-s − 64·95-s − 18·99-s − 24·101-s − 14·109-s + 31·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 9-s − 1.80·11-s + 3.67·19-s + 25-s + 0.742·29-s − 0.718·31-s + 3.74·41-s − 1.78·45-s + 3.23·55-s + 2.60·59-s − 1.57·79-s + 81-s − 1.69·89-s − 6.56·95-s − 1.80·99-s − 2.38·101-s − 1.34·109-s + 2.81·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962611822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962611822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 9 T^{2} - 208 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 185 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.31334669445838623682639990113, −7.10365631757318600519899018152, −7.00048018237222261760213757137, −6.54246563685918802109861910950, −6.38811216024287723597101720380, −5.69432002156047575343929288719, −5.67800944691663403298356849737, −5.59450076491160364774826823158, −5.52920260136622659280937084706, −5.02270565418853954589469780708, −4.78820270560852647865282149702, −4.66841974426817634879557763709, −4.17458753199840679756951925377, −4.15573363442531655335855105067, −3.82845209086311144048413634782, −3.71608156313355205832902244823, −3.16506432793313423237778978685, −3.04920154635995798618464664738, −2.72016682766894819974051636666, −2.67915935985273201602604406358, −2.10716521083837070371705689821, −1.68824620037295453690926620688, −1.02484058119904222543029692529, −0.966976187334345446841872998318, −0.42636961076067710366381340074,
0.42636961076067710366381340074, 0.966976187334345446841872998318, 1.02484058119904222543029692529, 1.68824620037295453690926620688, 2.10716521083837070371705689821, 2.67915935985273201602604406358, 2.72016682766894819974051636666, 3.04920154635995798618464664738, 3.16506432793313423237778978685, 3.71608156313355205832902244823, 3.82845209086311144048413634782, 4.15573363442531655335855105067, 4.17458753199840679756951925377, 4.66841974426817634879557763709, 4.78820270560852647865282149702, 5.02270565418853954589469780708, 5.52920260136622659280937084706, 5.59450076491160364774826823158, 5.67800944691663403298356849737, 5.69432002156047575343929288719, 6.38811216024287723597101720380, 6.54246563685918802109861910950, 7.00048018237222261760213757137, 7.10365631757318600519899018152, 7.31334669445838623682639990113
