| L(s) = 1 | − 8·5-s − 8·13-s + 24·17-s − 56·25-s + 128·29-s − 24·37-s + 160·41-s + 124·49-s + 48·53-s + 136·61-s + 64·65-s + 72·73-s − 192·85-s − 168·89-s + 104·97-s + 352·101-s + 48·109-s + 728·113-s + 568·121-s + 744·125-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 0.615·13-s + 1.41·17-s − 2.23·25-s + 4.41·29-s − 0.648·37-s + 3.90·41-s + 2.53·49-s + 0.905·53-s + 2.22·61-s + 0.984·65-s + 0.986·73-s − 2.25·85-s − 1.88·89-s + 1.07·97-s + 3.48·101-s + 0.440·109-s + 6.44·113-s + 4.69·121-s + 5.95·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.06·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.170148116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.170148116\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T + 52 T^{2} + 124 T^{3} + 1558 T^{4} + 124 p^{2} T^{5} + 52 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 124 T^{2} + 5002 T^{4} + 178832 T^{6} - 21308909 T^{8} + 178832 p^{4} T^{10} + 5002 p^{8} T^{12} - 124 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 568 T^{2} + 14804 p T^{4} - 31597576 T^{6} + 4462882054 T^{8} - 31597576 p^{4} T^{10} + 14804 p^{9} T^{12} - 568 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 4 T + 394 T^{2} - 272 T^{3} + 72883 T^{4} - 272 p^{2} T^{5} + 394 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 12 T + 436 T^{2} - 7380 T^{3} + 90294 T^{4} - 7380 p^{2} T^{5} + 436 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1340 T^{2} + 1130122 T^{4} - 629721776 T^{6} + 266027004115 T^{8} - 629721776 p^{4} T^{10} + 1130122 p^{8} T^{12} - 1340 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 2872 T^{2} + 3949276 T^{4} - 3455093896 T^{6} + 2139369590086 T^{8} - 3455093896 p^{4} T^{10} + 3949276 p^{8} T^{12} - 2872 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 64 T + 2596 T^{2} - 67264 T^{3} + 1985254 T^{4} - 67264 p^{2} T^{5} + 2596 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 184 p T^{2} + 15692188 T^{4} - 26845223416 T^{6} + 31031256945478 T^{8} - 26845223416 p^{4} T^{10} + 15692188 p^{8} T^{12} - 184 p^{13} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 12 T + 2650 T^{2} - 36432 T^{3} + 2846211 T^{4} - 36432 p^{2} T^{5} + 2650 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 80 T + 5956 T^{2} - 285680 T^{3} + 13442758 T^{4} - 285680 p^{2} T^{5} + 5956 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 12424 T^{2} + 71116252 T^{4} - 244515946552 T^{6} + 551660618273158 T^{8} - 244515946552 p^{4} T^{10} + 71116252 p^{8} T^{12} - 12424 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 2872 T^{2} + 15358300 T^{4} - 35747158408 T^{6} + 102474993504454 T^{8} - 35747158408 p^{4} T^{10} + 15358300 p^{8} T^{12} - 2872 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 24 T + 7204 T^{2} - 159624 T^{3} + 28315302 T^{4} - 159624 p^{2} T^{5} + 7204 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 11320 T^{2} + 50642716 T^{4} - 127137029128 T^{6} + 320662122219526 T^{8} - 127137029128 p^{4} T^{10} + 50642716 p^{8} T^{12} - 11320 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 68 T + 8266 T^{2} - 137072 T^{3} + 22167667 T^{4} - 137072 p^{2} T^{5} + 8266 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 27292 T^{2} + 345823978 T^{4} - 2700513303280 T^{6} + 14433873700614067 T^{8} - 2700513303280 p^{4} T^{10} + 345823978 p^{8} T^{12} - 27292 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 11080 T^{2} + 30779164 T^{4} + 199057866248 T^{6} - 1902994100012090 T^{8} + 199057866248 p^{4} T^{10} + 30779164 p^{8} T^{12} - 11080 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 36 T + 9130 T^{2} - 318096 T^{3} + 41836659 T^{4} - 318096 p^{2} T^{5} + 9130 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 41308 T^{2} + 786693994 T^{4} - 9045860371504 T^{6} + 68675576167144435 T^{8} - 9045860371504 p^{4} T^{10} + 786693994 p^{8} T^{12} - 41308 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 8200 T^{2} + 140721244 T^{4} - 1081724015032 T^{6} + 8821781005832710 T^{8} - 1081724015032 p^{4} T^{10} + 140721244 p^{8} T^{12} - 8200 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 84 T + 22900 T^{2} + 1903500 T^{3} + 237150582 T^{4} + 1903500 p^{2} T^{5} + 22900 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 52 T + 22522 T^{2} - 1223152 T^{3} + 279148675 T^{4} - 1223152 p^{2} T^{5} + 22522 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.68831540002152667137559650804, −3.68022076482309066831305093072, −3.51879626674420204093894745036, −3.44106177573650451902310343962, −3.31790903048030112448826236932, −3.01154351758696967411052925135, −2.98429995071267744654173145806, −2.74195701318936667850682720914, −2.70627186974806553608593646052, −2.57058222591157623448371248804, −2.45120203278259882416029388286, −2.40123787512670366782254058171, −2.09820652445766158426815065583, −2.02849608136922062627983733643, −1.98917915698573643982210310553, −1.72527206428198544922296210686, −1.52592501858461000623439795519, −1.32567585991287015201811595207, −0.956968111036679285865570597679, −0.923979759052119839117876720913, −0.876286958305806053852312984119, −0.66791169795716857855030019133, −0.62780421035023376824409747903, −0.40164512519473449376161214319, −0.17522947364113160425991031003,
0.17522947364113160425991031003, 0.40164512519473449376161214319, 0.62780421035023376824409747903, 0.66791169795716857855030019133, 0.876286958305806053852312984119, 0.923979759052119839117876720913, 0.956968111036679285865570597679, 1.32567585991287015201811595207, 1.52592501858461000623439795519, 1.72527206428198544922296210686, 1.98917915698573643982210310553, 2.02849608136922062627983733643, 2.09820652445766158426815065583, 2.40123787512670366782254058171, 2.45120203278259882416029388286, 2.57058222591157623448371248804, 2.70627186974806553608593646052, 2.74195701318936667850682720914, 2.98429995071267744654173145806, 3.01154351758696967411052925135, 3.31790903048030112448826236932, 3.44106177573650451902310343962, 3.51879626674420204093894745036, 3.68022076482309066831305093072, 3.68831540002152667137559650804
Plot not available for L-functions of degree greater than 10.
