| L(s) = 1 | + 16·7-s − 160·25-s + 112·31-s − 192·49-s + 160·73-s + 816·79-s + 36·81-s + 192·97-s + 976·103-s − 960·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.26e3·169-s + 173-s − 2.56e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 6.39·25-s + 3.61·31-s − 3.91·49-s + 2.19·73-s + 10.3·79-s + 4/9·81-s + 1.97·97-s + 9.47·103-s − 7.93·121-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 + 7.47·169-s + 0.00578·173-s − 14.6·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(175.7023282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(175.7023282\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 4 p^{2} T^{4} - 512 T^{6} + 470 p^{2} T^{8} - 512 p^{4} T^{10} - 4 p^{10} T^{12} + p^{16} T^{16} \) |
| good | 5 | \( ( 1 + 16 p T^{2} + 3004 T^{4} + 18672 p T^{6} + 2634054 T^{8} + 18672 p^{5} T^{10} + 3004 p^{8} T^{12} + 16 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 7 | \( ( 1 - 4 T + 88 T^{2} - 220 T^{3} + 3646 T^{4} - 220 p^{2} T^{5} + 88 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 11 | \( ( 1 + 480 T^{2} + 125212 T^{4} + 22531104 T^{6} + 3078264198 T^{8} + 22531104 p^{4} T^{10} + 125212 p^{8} T^{12} + 480 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 - 632 T^{2} + 214876 T^{4} - 54331336 T^{6} + 10620684806 T^{8} - 54331336 p^{4} T^{10} + 214876 p^{8} T^{12} - 632 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 - 1064 T^{2} + 654556 T^{4} - 282654744 T^{6} + 92440824774 T^{8} - 282654744 p^{4} T^{10} + 654556 p^{8} T^{12} - 1064 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 - 1184 T^{2} + 791836 T^{4} - 351676768 T^{6} + 134635543814 T^{8} - 351676768 p^{4} T^{10} + 791836 p^{8} T^{12} - 1184 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 - 1896 T^{2} + 2093020 T^{4} - 1543350744 T^{6} + 914302437702 T^{8} - 1543350744 p^{4} T^{10} + 2093020 p^{8} T^{12} - 1896 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 + 3024 T^{2} + 4188220 T^{4} + 3460574000 T^{6} + 2571955018182 T^{8} + 3460574000 p^{4} T^{10} + 4188220 p^{8} T^{12} + 3024 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 31 | \( ( 1 - 28 T + 3016 T^{2} - 63204 T^{3} + 3999966 T^{4} - 63204 p^{2} T^{5} + 3016 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 37 | \( ( 1 - 3640 T^{2} + 6795868 T^{4} - 12043846408 T^{6} + 19395045099910 T^{8} - 12043846408 p^{4} T^{10} + 6795868 p^{8} T^{12} - 3640 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 2472 T^{2} + 3015004 T^{4} - 2001918872 T^{6} + 1122734027334 T^{8} - 2001918872 p^{4} T^{10} + 3015004 p^{8} T^{12} - 2472 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 43 | \( ( 1 - 8992 T^{2} + 42471580 T^{4} - 131108267744 T^{6} + 285583273079942 T^{8} - 131108267744 p^{4} T^{10} + 42471580 p^{8} T^{12} - 8992 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 - 12296 T^{2} + 72283420 T^{4} - 268133584440 T^{6} + 697058148522822 T^{8} - 268133584440 p^{4} T^{10} + 72283420 p^{8} T^{12} - 12296 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 304 T^{2} + 25898428 T^{4} - 6393839056 T^{6} + 288293861465158 T^{8} - 6393839056 p^{4} T^{10} + 25898428 p^{8} T^{12} - 304 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 + 19008 T^{2} + 181109212 T^{4} + 1092163345344 T^{6} + 4537129454477574 T^{8} + 1092163345344 p^{4} T^{10} + 181109212 p^{8} T^{12} + 19008 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 19000 T^{2} + 184830940 T^{4} - 1158983771656 T^{6} + 5102256578203654 T^{8} - 1158983771656 p^{4} T^{10} + 184830940 p^{8} T^{12} - 19000 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 67 | \( ( 1 - 18112 T^{2} + 165933916 T^{4} - 1017402772288 T^{6} + 4957396059692422 T^{8} - 1017402772288 p^{4} T^{10} + 165933916 p^{8} T^{12} - 18112 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 17640 T^{2} + 102948316 T^{4} - 54971476568 T^{6} - 1448379553007034 T^{8} - 54971476568 p^{4} T^{10} + 102948316 p^{8} T^{12} - 17640 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 40 T + 15772 T^{2} - 446872 T^{3} + 110402182 T^{4} - 446872 p^{2} T^{5} + 15772 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 79 | \( ( 1 - 204 T + 32488 T^{2} - 3143092 T^{3} + 288852126 T^{4} - 3143092 p^{2} T^{5} + 32488 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 41696 T^{2} + 833769628 T^{4} + 10294236337952 T^{6} + 85511640874023430 T^{8} + 10294236337952 p^{4} T^{10} + 833769628 p^{8} T^{12} + 41696 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 17736 T^{2} + 318384412 T^{4} - 3335315895288 T^{6} + 31854516109881798 T^{8} - 3335315895288 p^{4} T^{10} + 318384412 p^{8} T^{12} - 17736 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 - 48 T + 29500 T^{2} - 1080784 T^{3} + 386488710 T^{4} - 1080784 p^{2} T^{5} + 29500 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
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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
−2.39505542920937389690844187901, −2.16631905985778602257032904381, −2.16592438102189621963084023411, −2.12281493820727465016326667012, −2.07305912916284957338629832474, −2.02140536178473440145989083306, −1.93449394414153898459687032933, −1.89600327911081781398091474918, −1.88285627471135063262706611869, −1.80194634552905201400799259707, −1.64742251008767348667556081070, −1.62932105856347638547369068200, −1.62759853644996990291887276136, −1.28083937404439999715592622840, −1.28038868023937209792498328264, −1.24765990110437050170836693269, −0.800364356728166449762481897366, −0.792855697823626883550754282544, −0.69619718895081540557598009873, −0.63984363524976836957656788662, −0.61371578092926373187402293455, −0.58941572965562794571843378193, −0.37397394063188883488233398943, −0.35987049773414664014622578292, −0.27858236843667850865111942747,
0.27858236843667850865111942747, 0.35987049773414664014622578292, 0.37397394063188883488233398943, 0.58941572965562794571843378193, 0.61371578092926373187402293455, 0.63984363524976836957656788662, 0.69619718895081540557598009873, 0.792855697823626883550754282544, 0.800364356728166449762481897366, 1.24765990110437050170836693269, 1.28038868023937209792498328264, 1.28083937404439999715592622840, 1.62759853644996990291887276136, 1.62932105856347638547369068200, 1.64742251008767348667556081070, 1.80194634552905201400799259707, 1.88285627471135063262706611869, 1.89600327911081781398091474918, 1.93449394414153898459687032933, 2.02140536178473440145989083306, 2.07305912916284957338629832474, 2.12281493820727465016326667012, 2.16592438102189621963084023411, 2.16631905985778602257032904381, 2.39505542920937389690844187901
Plot not available for L-functions of degree greater than 10.
