| L(s) = 1 | + 4-s + 2·7-s − 8·13-s + 16-s − 5·25-s + 2·28-s + 4·31-s − 20·37-s − 10·43-s − 9·49-s − 8·52-s + 14·61-s + 5·64-s + 16·67-s + 14·73-s + 16·79-s − 16·91-s + 16·97-s − 5·100-s + 52·103-s − 44·109-s + 2·112-s − 17·121-s + 4·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s − 2.21·13-s + 1/4·16-s − 25-s + 0.377·28-s + 0.718·31-s − 3.28·37-s − 1.52·43-s − 9/7·49-s − 1.10·52-s + 1.79·61-s + 5/8·64-s + 1.95·67-s + 1.63·73-s + 1.80·79-s − 1.67·91-s + 1.62·97-s − 1/2·100-s + 5.12·103-s − 4.21·109-s + 0.188·112-s − 1.54·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2684213829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2684213829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + p T^{2} + 48 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 17 T^{2} + 240 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 53 T^{2} + 1272 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 2310 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 128 T^{2} + 7326 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 89 T^{2} + 5400 T^{4} + 89 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} + 5286 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 6966 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 + 92 T^{2} + 3750 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 188 T^{2} + 17862 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 116 T^{2} + 18678 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.28140413603557895549266656194, −6.07079224679349013554140206466, −5.45920783101604640756852161295, −5.39792397668946556413082770959, −5.28212705511001476648400284583, −5.12388512867944117893568830528, −4.96885604484781853550300728028, −4.83122173983688967369593944988, −4.70777268045945160696053442405, −4.18056684703187544539503093712, −4.07629784096682372827426728675, −3.84691705983557513712187917272, −3.49990123584584951392162247117, −3.45111956918923892243343860378, −3.27390820371099187542564429113, −2.90063562610421070142928755133, −2.57029133115327689158214690480, −2.32678951569004499486220857304, −2.20153956637204838197659684097, −1.93293093912848200396179835667, −1.72824266074758839511972517124, −1.57182415139342945568633083366, −0.840618963060323006620956229547, −0.795768339598629908094143678954, −0.07821443676109723614583273857,
0.07821443676109723614583273857, 0.795768339598629908094143678954, 0.840618963060323006620956229547, 1.57182415139342945568633083366, 1.72824266074758839511972517124, 1.93293093912848200396179835667, 2.20153956637204838197659684097, 2.32678951569004499486220857304, 2.57029133115327689158214690480, 2.90063562610421070142928755133, 3.27390820371099187542564429113, 3.45111956918923892243343860378, 3.49990123584584951392162247117, 3.84691705983557513712187917272, 4.07629784096682372827426728675, 4.18056684703187544539503093712, 4.70777268045945160696053442405, 4.83122173983688967369593944988, 4.96885604484781853550300728028, 5.12388512867944117893568830528, 5.28212705511001476648400284583, 5.39792397668946556413082770959, 5.45920783101604640756852161295, 6.07079224679349013554140206466, 6.28140413603557895549266656194
