| L(s) = 1 | − 3·3-s + 4·5-s + 5·7-s + 5·9-s − 5·11-s − 6·13-s − 12·15-s + 10·17-s − 15·21-s − 2·23-s + 10·25-s − 6·27-s + 2·29-s + 4·31-s + 15·33-s + 20·35-s + 4·37-s + 18·39-s + 29·41-s − 4·43-s + 20·45-s + 9·47-s − 49-s − 30·51-s − 3·53-s − 20·55-s + 10·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 1.88·7-s + 5/3·9-s − 1.50·11-s − 1.66·13-s − 3.09·15-s + 2.42·17-s − 3.27·21-s − 0.417·23-s + 2·25-s − 1.15·27-s + 0.371·29-s + 0.718·31-s + 2.61·33-s + 3.38·35-s + 0.657·37-s + 2.88·39-s + 4.52·41-s − 0.609·43-s + 2.98·45-s + 1.31·47-s − 1/7·49-s − 4.20·51-s − 0.412·53-s − 2.69·55-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.299352238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.299352238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + p T + 4 T^{2} + p T^{3} + 4 T^{4} + p^{2} T^{5} + 4 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 5 T + 26 T^{2} - 93 T^{3} + 272 T^{4} - 93 p T^{5} + 26 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 5 T + 12 T^{2} - 27 T^{3} - 218 T^{4} - 27 p T^{5} + 12 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 6 T + 32 T^{2} + 150 T^{3} + 654 T^{4} + 150 p T^{5} + 32 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 10 T + 72 T^{2} - 402 T^{3} + 1918 T^{4} - 402 p T^{5} + 72 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 p T^{2} + 54 T^{3} + 786 T^{4} + 54 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 48 T^{2} + 162 T^{3} + 1342 T^{4} + 162 p T^{5} + 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 72 T^{2} - 198 T^{3} + 2686 T^{4} - 198 p T^{5} + 72 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 4 T + 34 T^{2} - 22 T^{3} + 334 T^{4} - 22 p T^{5} + 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 29 T + 438 T^{2} - 4395 T^{3} + 32386 T^{4} - 4395 p T^{5} + 438 p^{2} T^{6} - 29 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 4 T + 44 T^{2} + 228 T^{3} + 1382 T^{4} + 228 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 9 T + 86 T^{2} - 549 T^{3} + 3720 T^{4} - 549 p T^{5} + 86 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 3 T + 158 T^{2} + 513 T^{3} + 11202 T^{4} + 513 p T^{5} + 158 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 10 T + 174 T^{2} - 924 T^{3} + 11602 T^{4} - 924 p T^{5} + 174 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 14 T + 274 T^{2} + 2792 T^{3} + 27730 T^{4} + 2792 p T^{5} + 274 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 17 T + 276 T^{2} - 3381 T^{3} + 28822 T^{4} - 3381 p T^{5} + 276 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 7 T + 2 p T^{2} - 1185 T^{3} + 11018 T^{4} - 1185 p T^{5} + 2 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 24 T + 494 T^{2} - 6042 T^{3} + 65238 T^{4} - 6042 p T^{5} + 494 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 31 T + 576 T^{2} + 7431 T^{3} + 76600 T^{4} + 7431 p T^{5} + 576 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 4 T + 228 T^{2} + 828 T^{3} + 28246 T^{4} + 828 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 12 T + 308 T^{2} - 2820 T^{3} + 41334 T^{4} - 2820 p T^{5} + 308 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.16335259409152204067191248031, −5.69976754791716072902570141097, −5.69006620924904598308902070904, −5.64419739588587604891787412058, −5.42291545901554954663548117262, −5.22583676194416762711093747779, −5.11619814813277435969066386911, −4.83424223679118025844607832366, −4.74339821399949667739476278709, −4.60801675891498516116129207317, −4.11100564886281731391978783997, −4.06163134422965226458281426861, −3.95441929132669838458699819172, −3.37380411819933157047651251243, −3.03629939046392390103484718530, −2.83055028271951690415837524828, −2.66153156867704041960025321639, −2.36126271062246058171525404405, −2.13614557423074112448705269769, −2.09881113084360961475410794461, −1.48532616280680045334356300557, −1.34393085003879774748073575820, −0.901018545996467818620558522561, −0.897787802618397102339712087190, −0.36915076010304794162596319227,
0.36915076010304794162596319227, 0.897787802618397102339712087190, 0.901018545996467818620558522561, 1.34393085003879774748073575820, 1.48532616280680045334356300557, 2.09881113084360961475410794461, 2.13614557423074112448705269769, 2.36126271062246058171525404405, 2.66153156867704041960025321639, 2.83055028271951690415837524828, 3.03629939046392390103484718530, 3.37380411819933157047651251243, 3.95441929132669838458699819172, 4.06163134422965226458281426861, 4.11100564886281731391978783997, 4.60801675891498516116129207317, 4.74339821399949667739476278709, 4.83424223679118025844607832366, 5.11619814813277435969066386911, 5.22583676194416762711093747779, 5.42291545901554954663548117262, 5.64419739588587604891787412058, 5.69006620924904598308902070904, 5.69976754791716072902570141097, 6.16335259409152204067191248031
