| L(s) = 1 | − 8·4-s − 4·5-s + 2·7-s − 2·9-s − 10·11-s + 40·16-s + 2·17-s + 32·20-s + 4·23-s + 10·25-s − 16·28-s − 8·35-s + 16·36-s − 34·43-s + 80·44-s + 8·45-s − 8·47-s − 23·49-s + 40·55-s − 20·61-s − 4·63-s − 160·64-s − 16·68-s − 44·73-s − 20·77-s − 160·80-s + 5·81-s + ⋯ |
| L(s) = 1 | − 4·4-s − 1.78·5-s + 0.755·7-s − 2/3·9-s − 3.01·11-s + 10·16-s + 0.485·17-s + 7.15·20-s + 0.834·23-s + 2·25-s − 3.02·28-s − 1.35·35-s + 8/3·36-s − 5.18·43-s + 12.0·44-s + 1.19·45-s − 1.16·47-s − 3.28·49-s + 5.39·55-s − 2.56·61-s − 0.503·63-s − 20·64-s − 1.94·68-s − 5.14·73-s − 2.27·77-s − 17.8·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 2 T^{2} - T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 42 T^{2} + 759 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 96 T^{2} + 3966 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 74 T^{2} + 2791 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 18 T^{2} + 2799 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 6 T^{2} + 951 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 17 T + 157 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 27 T^{2} + 1449 T^{4} + 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 211 T^{2} + 18061 T^{4} + 211 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 203 T^{2} + 18829 T^{4} + 203 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 114 T^{2} + 6111 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 134 T^{2} + 15351 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 137 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 111 T^{2} + 15921 T^{4} + 111 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 258 T^{2} + 35439 T^{4} + 258 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.24925219821629660566955284599, −6.91591456807867376754611517098, −6.42676314708237240886196993022, −6.30600223120693907706749402635, −6.20446574084068106303835287022, −5.61490689202384684115410104138, −5.49580290401254317588508523811, −5.42571300836847964032064170405, −5.17802306067682585821016884206, −4.83964967599617408688387679193, −4.77783931152790934855537850894, −4.76023683760472086402988622022, −4.72897109025186701158251897407, −4.34384199858207447852723073499, −3.94236001789148049010090535877, −3.77657602642235815302396227066, −3.55586122555851354197136028429, −3.07140620879446014618282527088, −3.04471750615279069127583911746, −3.03075989187154571245610161709, −2.97490844159120497403468506804, −1.97247302592756378320630139447, −1.57392083943964158900590007818, −1.35422923237514510654852541383, −0.988668033490779323044994557794, 0, 0, 0, 0,
0.988668033490779323044994557794, 1.35422923237514510654852541383, 1.57392083943964158900590007818, 1.97247302592756378320630139447, 2.97490844159120497403468506804, 3.03075989187154571245610161709, 3.04471750615279069127583911746, 3.07140620879446014618282527088, 3.55586122555851354197136028429, 3.77657602642235815302396227066, 3.94236001789148049010090535877, 4.34384199858207447852723073499, 4.72897109025186701158251897407, 4.76023683760472086402988622022, 4.77783931152790934855537850894, 4.83964967599617408688387679193, 5.17802306067682585821016884206, 5.42571300836847964032064170405, 5.49580290401254317588508523811, 5.61490689202384684115410104138, 6.20446574084068106303835287022, 6.30600223120693907706749402635, 6.42676314708237240886196993022, 6.91591456807867376754611517098, 7.24925219821629660566955284599
