| L(s) = 1 | + 24·9-s − 96·11-s + 12·25-s + 64·29-s + 112·49-s + 384·71-s + 608·79-s + 324·81-s − 2.30e3·99-s − 112·109-s + 3.71e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 616·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 8.72·11-s + 0.479·25-s + 2.20·29-s + 16/7·49-s + 5.40·71-s + 7.69·79-s + 4·81-s − 23.2·99-s − 1.02·109-s + 30.6·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.64·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.181541417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.181541417\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{8} \) |
| 5 | \( 1 - 12 T^{2} - 376 T^{4} - 132 p^{2} T^{6} + 942 p^{4} T^{8} - 132 p^{6} T^{10} - 376 p^{8} T^{12} - 12 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( 1 - 16 p T^{2} + 3932 T^{4} - 24720 T^{6} - 24154 p^{2} T^{8} - 24720 p^{4} T^{10} + 3932 p^{8} T^{12} - 16 p^{13} T^{14} + p^{16} T^{16} \) |
| good | 11 | \( ( 1 + 24 T + 512 T^{2} + 6936 T^{3} + 89778 T^{4} + 6936 p^{2} T^{5} + 512 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 13 | \( ( 1 + 308 T^{2} + 54824 T^{4} + 3243420 T^{6} - 106725490 T^{8} + 3243420 p^{4} T^{10} + 54824 p^{8} T^{12} + 308 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 + 1316 T^{2} + 51880 p T^{4} + 398466540 T^{6} + 132906424526 T^{8} + 398466540 p^{4} T^{10} + 51880 p^{9} T^{12} + 1316 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 - 1092 T^{2} + 576680 T^{4} - 222586380 T^{6} + 81331407246 T^{8} - 222586380 p^{4} T^{10} + 576680 p^{8} T^{12} - 1092 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 - 1300 T^{2} + 1274456 T^{4} - 947494620 T^{6} + 528617435630 T^{8} - 947494620 p^{4} T^{10} + 1274456 p^{8} T^{12} - 1300 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 16 T + 44 p T^{2} - 55408 T^{3} + 751270 T^{4} - 55408 p^{2} T^{5} + 44 p^{5} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 3948 T^{2} + 5903432 T^{4} - 3942646212 T^{6} + 1976573418510 T^{8} - 3942646212 p^{4} T^{10} + 5903432 p^{8} T^{12} - 3948 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 - 2416 T^{2} + 5835932 T^{4} - 9959371920 T^{6} + 14384225705606 T^{8} - 9959371920 p^{4} T^{10} + 5835932 p^{8} T^{12} - 2416 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 12516 T^{2} + 70001768 T^{4} - 228356671020 T^{6} + 475054496880078 T^{8} - 228356671020 p^{4} T^{10} + 70001768 p^{8} T^{12} - 12516 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 43 | \( ( 1 - 1912 T^{2} + 2635964 T^{4} - 3280241736 T^{6} + 4054697588678 T^{8} - 3280241736 p^{4} T^{10} + 2635964 p^{8} T^{12} - 1912 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 144 p T^{2} + 28283420 T^{4} + 81813152400 T^{6} + 198252999971526 T^{8} + 81813152400 p^{4} T^{10} + 28283420 p^{8} T^{12} + 144 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 6868 T^{2} + 39779480 T^{4} - 162177493980 T^{6} + 178595339294 p^{2} T^{8} - 162177493980 p^{4} T^{10} + 39779480 p^{8} T^{12} - 6868 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 11736 T^{2} + 79787516 T^{4} - 412369170408 T^{6} + 1633889740528710 T^{8} - 412369170408 p^{4} T^{10} + 79787516 p^{8} T^{12} - 11736 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 15944 T^{2} + 96425692 T^{4} - 269693317496 T^{6} + 596407807846918 T^{8} - 269693317496 p^{4} T^{10} + 96425692 p^{8} T^{12} - 15944 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 67 | \( ( 1 - 18688 T^{2} + 141351740 T^{4} - 579001624320 T^{6} + 2070725817126086 T^{8} - 579001624320 p^{4} T^{10} + 141351740 p^{8} T^{12} - 18688 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 96 T + 14048 T^{2} - 646080 T^{3} + 72093138 T^{4} - 646080 p^{2} T^{5} + 14048 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 73 | \( ( 1 + 18612 T^{2} + 177429992 T^{4} + 1152928560348 T^{6} + 6336699932034510 T^{8} + 1152928560348 p^{4} T^{10} + 177429992 p^{8} T^{12} + 18612 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 152 T + 28916 T^{2} - 2541000 T^{3} + 270417254 T^{4} - 2541000 p^{2} T^{5} + 28916 p^{4} T^{6} - 152 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 39632 T^{2} + 756012572 T^{4} + 9074582537520 T^{6} + 74694570513655814 T^{8} + 9074582537520 p^{4} T^{10} + 756012572 p^{8} T^{12} + 39632 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 38508 T^{2} + 778838024 T^{4} - 10318651040964 T^{6} + 96273467949842958 T^{8} - 10318651040964 p^{4} T^{10} + 778838024 p^{8} T^{12} - 38508 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 + 54804 T^{2} + 1439825768 T^{4} + 23691050976252 T^{6} + 266516143722806094 T^{8} + 23691050976252 p^{4} T^{10} + 1439825768 p^{8} T^{12} + 54804 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.16863778712383023964542489527, −2.09750309789678971930530216371, −2.00400269932028868961032284001, −1.96538496068151063797698543859, −1.95647680729935982419902387024, −1.94496146136360218088827074438, −1.66398878552824104722908124353, −1.66197267618853995059771616562, −1.50081897638766491185143519390, −1.42390994811726544164634309579, −1.28652670315891422454899804113, −1.18723306870460044712231250292, −1.13005554872866252270083971073, −1.03052083355374575335962371775, −0.986950602010611349370418257089, −0.890825454789922708726662814747, −0.886638663901407371578952441253, −0.77772974598795894659720197127, −0.48874222666876348264376502095, −0.45934747344862667804023503144, −0.43980174685380472384976709935, −0.41206618499320446754515777054, −0.33361188332919040386831902421, −0.17037775170297148981261111819, −0.10974613594449472204017978065,
0.10974613594449472204017978065, 0.17037775170297148981261111819, 0.33361188332919040386831902421, 0.41206618499320446754515777054, 0.43980174685380472384976709935, 0.45934747344862667804023503144, 0.48874222666876348264376502095, 0.77772974598795894659720197127, 0.886638663901407371578952441253, 0.890825454789922708726662814747, 0.986950602010611349370418257089, 1.03052083355374575335962371775, 1.13005554872866252270083971073, 1.18723306870460044712231250292, 1.28652670315891422454899804113, 1.42390994811726544164634309579, 1.50081897638766491185143519390, 1.66197267618853995059771616562, 1.66398878552824104722908124353, 1.94496146136360218088827074438, 1.95647680729935982419902387024, 1.96538496068151063797698543859, 2.00400269932028868961032284001, 2.09750309789678971930530216371, 2.16863778712383023964542489527
Plot not available for L-functions of degree greater than 10.
