| L(s) = 1 | − 4·2-s + 2·3-s + 11·4-s + 4·5-s − 8·6-s − 22·8-s − 9-s − 16·10-s + 2·11-s + 22·12-s − 8·13-s + 8·15-s + 36·16-s − 4·17-s + 4·18-s − 2·19-s + 44·20-s − 8·22-s − 4·23-s − 44·24-s + 5·25-s + 32·26-s − 2·27-s − 32·30-s + 12·31-s − 52·32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.15·3-s + 11/2·4-s + 1.78·5-s − 3.26·6-s − 7.77·8-s − 1/3·9-s − 5.05·10-s + 0.603·11-s + 6.35·12-s − 2.21·13-s + 2.06·15-s + 9·16-s − 0.970·17-s + 0.942·18-s − 0.458·19-s + 9.83·20-s − 1.70·22-s − 0.834·23-s − 8.98·24-s + 25-s + 6.27·26-s − 0.384·27-s − 5.84·30-s + 2.15·31-s − 9.19·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9081410578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9081410578\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + 5 T^{2} - p T^{3} - 11 T^{4} - p^{2} T^{5} + 5 p^{2} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 100 T^{3} + 271 T^{4} + 100 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 4 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 244 T^{3} + 1588 T^{4} + 244 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 288 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 528 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 157 T^{2} - 1308 T^{3} + 11088 T^{4} - 1308 p T^{5} + 157 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 834 p T^{5} + 137 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 144 T^{2} + 24 p T^{3} - 31057 T^{4} + 24 p^{2} T^{5} + 144 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 140 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.938598668038996038116916410478, −8.475168000469860911523449851948, −8.225338497739255309866223726014, −8.200595171564925584220776174621, −8.120476242988960430697599433712, −7.70027515690924379740748435776, −7.25128396601842605418829092890, −7.17122927069378958133034539429, −6.77622377285187602716424416509, −6.67722601813082860408384906143, −6.26303189106745977186327868055, −6.08106135955873524882917946456, −5.98975151675537099346997134530, −5.35017392651604571999591095560, −5.00609288249513974887296413336, −4.59663359487590538473910174108, −4.46865246690969969199442214683, −3.62737802420282452233745761757, −3.19843128648142824645905099243, −2.83583777330632344433760511590, −2.34402929176256115520639832499, −2.19683404618992609901026736825, −2.13547553519354470592275621791, −1.61174520549788935873907986488, −0.71389620344308966359474613873,
0.71389620344308966359474613873, 1.61174520549788935873907986488, 2.13547553519354470592275621791, 2.19683404618992609901026736825, 2.34402929176256115520639832499, 2.83583777330632344433760511590, 3.19843128648142824645905099243, 3.62737802420282452233745761757, 4.46865246690969969199442214683, 4.59663359487590538473910174108, 5.00609288249513974887296413336, 5.35017392651604571999591095560, 5.98975151675537099346997134530, 6.08106135955873524882917946456, 6.26303189106745977186327868055, 6.67722601813082860408384906143, 6.77622377285187602716424416509, 7.17122927069378958133034539429, 7.25128396601842605418829092890, 7.70027515690924379740748435776, 8.120476242988960430697599433712, 8.200595171564925584220776174621, 8.225338497739255309866223726014, 8.475168000469860911523449851948, 8.938598668038996038116916410478
