| L(s) = 1 | + 8·2-s − 6·3-s + 40·4-s − 4·5-s − 48·6-s + 160·8-s + 13·9-s − 32·10-s − 240·12-s − 12·13-s + 24·15-s + 560·16-s − 32·17-s + 104·18-s + 42·19-s − 160·20-s + 12·23-s − 960·24-s + 24·25-s − 96·26-s − 6·27-s − 4·29-s + 192·30-s + 1.79e3·32-s − 256·34-s + 520·36-s − 96·37-s + ⋯ |
| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s − 4/5·5-s − 8·6-s + 20·8-s + 13/9·9-s − 3.19·10-s − 20·12-s − 0.923·13-s + 8/5·15-s + 35·16-s − 1.88·17-s + 52/9·18-s + 2.21·19-s − 8·20-s + 0.521·23-s − 40·24-s + 0.959·25-s − 3.69·26-s − 2/9·27-s − 0.137·29-s + 32/5·30-s + 56·32-s − 7.52·34-s + 14.4·36-s − 2.59·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.40565942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.40565942\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 19 | $C_2^2$ | \( 1 - 42 T + 47 p T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + 23 T^{2} + 22 p T^{3} + 148 T^{4} + 22 p^{3} T^{5} + 23 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 104 T^{3} - 449 T^{4} - 104 p^{2} T^{5} - 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 4482 T^{4} - 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 p T^{2} + 26403 T^{4} + 2 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T - 133 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 32 T + 310 T^{2} + 256 p T^{3} + 451 p^{2} T^{4} + 256 p^{3} T^{5} + 310 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 308 T^{2} - 3120 T^{3} - 186849 T^{4} - 3120 p^{2} T^{5} + 308 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 1640 T^{2} - 104 T^{3} + 2021599 T^{4} - 104 p^{2} T^{5} - 1640 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2264 T^{2} + 2540466 T^{4} - 2264 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 48 T + 2564 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 26 T - 2375 T^{2} + 8086 T^{3} + 5346484 T^{4} + 8086 p^{2} T^{5} - 2375 p^{4} T^{6} - 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 72 T + 4858 T^{2} + 225360 T^{3} + 9573171 T^{4} + 225360 p^{2} T^{5} + 4858 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3928 T^{2} + 10549503 T^{4} + 3928 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 124 T + 6394 T^{2} - 417136 T^{3} + 29594659 T^{4} - 417136 p^{2} T^{5} + 6394 p^{4} T^{6} - 124 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 78 T + 9487 T^{2} + 581802 T^{3} + 50578788 T^{4} + 581802 p^{2} T^{5} + 9487 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 44 T - 5960 T^{2} + 19976 T^{3} + 41297119 T^{4} + 19976 p^{2} T^{5} - 5960 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 102 T + 8903 T^{2} + 554370 T^{3} + 24955956 T^{4} + 554370 p^{2} T^{5} + 8903 p^{4} T^{6} + 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 204 T + 25462 T^{2} - 2364360 T^{3} + 178845171 T^{4} - 2364360 p^{2} T^{5} + 25462 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 26 T - 9671 T^{2} - 8086 T^{3} + 75059764 T^{4} - 8086 p^{2} T^{5} - 9671 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 360 T + 65842 T^{2} + 8151120 T^{3} + 743321283 T^{4} + 8151120 p^{2} T^{5} + 65842 p^{4} T^{6} + 360 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 18170 T^{2} + 161815347 T^{4} - 18170 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 16 T - 1130 T^{2} - 231296 T^{3} - 62849021 T^{4} - 231296 p^{2} T^{5} - 1130 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 234 T + 22369 T^{2} + 3175146 T^{3} + 445189284 T^{4} + 3175146 p^{2} T^{5} + 22369 p^{4} T^{6} + 234 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−10.95768244825309701870229730382, −10.59511886577952237863530976342, −10.33914208074678793775420571991, −10.16613175371278255627610311478, −9.599055422352761635421039112881, −9.049295005627654440849715377758, −8.319770056994913734956758084039, −8.220506355573729991745796987706, −7.46584176149451481536715693669, −7.20492768639750518730099848790, −7.08132331610636919852417243452, −6.82585109614242021533208711003, −6.67959576744378280621532437844, −5.95838159015304368150552002255, −5.77623636050006051218960637184, −5.42796947931396760924701382771, −5.26597033623146415118511824267, −4.98151321185819561980978028228, −4.41314192393084445856668560605, −4.37251911802085148314727859887, −3.83287573508576673010953296209, −3.11555682435495311365076196549, −3.03558867566400386420852005128, −2.28454488513586563838736527553, −1.42021795205647615767728878716,
1.42021795205647615767728878716, 2.28454488513586563838736527553, 3.03558867566400386420852005128, 3.11555682435495311365076196549, 3.83287573508576673010953296209, 4.37251911802085148314727859887, 4.41314192393084445856668560605, 4.98151321185819561980978028228, 5.26597033623146415118511824267, 5.42796947931396760924701382771, 5.77623636050006051218960637184, 5.95838159015304368150552002255, 6.67959576744378280621532437844, 6.82585109614242021533208711003, 7.08132331610636919852417243452, 7.20492768639750518730099848790, 7.46584176149451481536715693669, 8.220506355573729991745796987706, 8.319770056994913734956758084039, 9.049295005627654440849715377758, 9.599055422352761635421039112881, 10.16613175371278255627610311478, 10.33914208074678793775420571991, 10.59511886577952237863530976342, 10.95768244825309701870229730382
