| L(s) = 1 | + 6·3-s − 6·5-s − 8·7-s + 9·9-s − 18·11-s − 36·15-s − 30·17-s − 6·19-s − 48·21-s − 30·23-s − 5·25-s − 18·27-s + 48·29-s + 42·31-s − 108·33-s + 48·35-s − 62·37-s + 8·43-s − 54·45-s − 174·47-s + 22·49-s − 180·51-s − 78·53-s + 108·55-s − 36·57-s + 78·59-s − 42·61-s + ⋯ |
| L(s) = 1 | + 2·3-s − 6/5·5-s − 8/7·7-s + 9-s − 1.63·11-s − 2.39·15-s − 1.76·17-s − 0.315·19-s − 2.28·21-s − 1.30·23-s − 1/5·25-s − 2/3·27-s + 1.65·29-s + 1.35·31-s − 3.27·33-s + 1.37·35-s − 1.67·37-s + 8/43·43-s − 6/5·45-s − 3.70·47-s + 0.448·49-s − 3.52·51-s − 1.47·53-s + 1.96·55-s − 0.631·57-s + 1.32·59-s − 0.688·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4194734070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4194734070\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 8 T + 6 p T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T + p^{3} T^{2} - 10 p^{2} T^{3} + 28 p^{2} T^{4} - 10 p^{4} T^{5} + p^{7} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2}( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 18 T + 19 T^{2} + 1134 T^{3} + 39180 T^{4} + 1134 p^{2} T^{5} + 19 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 30 T + 929 T^{2} + 1110 p T^{3} + 1380 p^{2} T^{4} + 1110 p^{3} T^{5} + 929 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 731 T^{2} + 4314 T^{3} + 390972 T^{4} + 4314 p^{2} T^{5} + 731 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 30 T - 221 T^{2} + 1890 T^{3} + 500700 T^{4} + 1890 p^{2} T^{5} - 221 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 1754 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 42 T + 1307 T^{2} - 30198 T^{3} + 158508 T^{4} - 30198 p^{2} T^{5} + 1307 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 11842 p^{2} T^{5} + 1297 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 3630 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 174 T + 17027 T^{2} + 1206690 T^{3} + 65507772 T^{4} + 1206690 p^{2} T^{5} + 17027 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 78 T - 767 T^{2} + 96174 T^{3} + 21955764 T^{4} + 96174 p^{2} T^{5} - 767 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 78 T + 5747 T^{2} - 290082 T^{3} + 8773068 T^{4} - 290082 p^{2} T^{5} + 5747 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 42 T + 2033 T^{2} + 60690 T^{3} - 9569868 T^{4} + 60690 p^{2} T^{5} + 2033 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 58 T - 2405 T^{2} + 186122 T^{3} - 1970756 T^{4} + 186122 p^{2} T^{5} - 2405 p^{4} T^{6} - 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 8318 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 318 T + 51257 T^{2} - 5580582 T^{3} + 459199092 T^{4} - 5580582 p^{2} T^{5} + 51257 p^{4} T^{6} - 318 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 110 T - 2957 T^{2} + 283250 T^{3} + 112247068 T^{4} + 283250 p^{2} T^{5} - 2957 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 378 T + 71921 T^{2} + 9182754 T^{3} + 904668996 T^{4} + 9182754 p^{2} T^{5} + 71921 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−9.853984894368964302277428589913, −9.553236894118101516672293665467, −9.321854854880656996928390809260, −8.589768597265351541469118858960, −8.430553425749886925948109958511, −8.332779201133067756524802829264, −8.257914031204176219385176041721, −8.135645518052941304257035704250, −7.68203663703026323850118543243, −7.01483914921713818395262592519, −6.99283815459934172512038125282, −6.70619055757387246125579062869, −6.25332705936842072412796603976, −5.99952346166238881135620507230, −5.42792241743296705012890498483, −4.98651900831323812093226530624, −4.63601799563782964080174216972, −4.26478411159090463935034446745, −3.90016960281137215498826958106, −3.24551277998286156013902977159, −3.14778055764061173912277921114, −2.93265331128145043796726622751, −2.16854593935871317076830429095, −2.07087551694015619277399623509, −0.22827362820321006450031382378,
0.22827362820321006450031382378, 2.07087551694015619277399623509, 2.16854593935871317076830429095, 2.93265331128145043796726622751, 3.14778055764061173912277921114, 3.24551277998286156013902977159, 3.90016960281137215498826958106, 4.26478411159090463935034446745, 4.63601799563782964080174216972, 4.98651900831323812093226530624, 5.42792241743296705012890498483, 5.99952346166238881135620507230, 6.25332705936842072412796603976, 6.70619055757387246125579062869, 6.99283815459934172512038125282, 7.01483914921713818395262592519, 7.68203663703026323850118543243, 8.135645518052941304257035704250, 8.257914031204176219385176041721, 8.332779201133067756524802829264, 8.430553425749886925948109958511, 8.589768597265351541469118858960, 9.321854854880656996928390809260, 9.553236894118101516672293665467, 9.853984894368964302277428589913
