| L(s) = 1 | + 8·4-s + 52·16-s − 64·29-s + 32·41-s − 408·49-s + 272·61-s + 240·64-s + 256·89-s − 224·101-s + 1.04e3·109-s − 512·116-s + 528·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 256·164-s + 167-s + 1.00e3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·4-s + 13/4·16-s − 2.20·29-s + 0.780·41-s − 8.32·49-s + 4.45·61-s + 15/4·64-s + 2.87·89-s − 2.21·101-s + 9.54·109-s − 4.41·116-s + 4.36·121-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 + 1.56·164-s + 0.00598·167-s + 5.91·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3345051158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3345051158\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T^{2} + 3 p^{2} T^{4} + 5 p^{4} T^{6} - 31 p^{4} T^{8} + 5 p^{8} T^{10} + 3 p^{10} T^{12} - p^{15} T^{14} + p^{16} T^{16} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 204 T^{2} + 18426 T^{4} + 1076112 T^{6} + 53449475 T^{8} + 1076112 p^{4} T^{10} + 18426 p^{8} T^{12} + 204 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 11 | \( ( 1 - 24 p T^{2} + 33852 T^{4} - 3268920 T^{6} + 337315526 T^{8} - 3268920 p^{4} T^{10} + 33852 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 - 500 T^{2} + 138714 T^{4} - 168304 p^{2} T^{6} + 4888018019 T^{8} - 168304 p^{6} T^{10} + 138714 p^{8} T^{12} - 500 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 - 648 T^{2} + 410236 T^{4} - 149339448 T^{6} + 52652093574 T^{8} - 149339448 p^{4} T^{10} + 410236 p^{8} T^{12} - 648 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 - 1756 T^{2} + 1510026 T^{4} - 44288912 p T^{6} + 346336758035 T^{8} - 44288912 p^{5} T^{10} + 1510026 p^{8} T^{12} - 1756 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 + 2248 T^{2} + 2472828 T^{4} + 1852675064 T^{6} + 1086691824134 T^{8} + 1852675064 p^{4} T^{10} + 2472828 p^{8} T^{12} + 2248 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 + 16 T + 1764 T^{2} + 10544 T^{3} + 1646102 T^{4} + 10544 p^{2} T^{5} + 1764 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 2028 T^{2} + 2805402 T^{4} - 2884712400 T^{6} + 3076391251619 T^{8} - 2884712400 p^{4} T^{10} + 2805402 p^{8} T^{12} - 2028 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 - 1784 T^{2} + 7604700 T^{4} - 9917137480 T^{6} + 21466335785606 T^{8} - 9917137480 p^{4} T^{10} + 7604700 p^{8} T^{12} - 1784 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 8 T + 1756 T^{2} - 78008 T^{3} + 3756598 T^{4} - 78008 p^{2} T^{5} + 1756 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 + 7900 T^{2} + 33132810 T^{4} + 96255124016 T^{6} + 206878197918227 T^{8} + 96255124016 p^{4} T^{10} + 33132810 p^{8} T^{12} + 7900 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 5656 T^{2} + 20883420 T^{4} + 53920616360 T^{6} + 122334534618566 T^{8} + 53920616360 p^{4} T^{10} + 20883420 p^{8} T^{12} + 5656 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 8504 T^{2} + 37649340 T^{4} - 110228557960 T^{6} + 302042397010886 T^{8} - 110228557960 p^{4} T^{10} + 37649340 p^{8} T^{12} - 8504 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 21656 T^{2} + 220430236 T^{4} - 1377870886568 T^{6} + 5784129582348550 T^{8} - 1377870886568 p^{4} T^{10} + 220430236 p^{8} T^{12} - 21656 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 68 T + 6786 T^{2} - 434848 T^{3} + 32554859 T^{4} - 434848 p^{2} T^{5} + 6786 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 + 14364 T^{2} + 117868746 T^{4} + 700523495472 T^{6} + 3348544470504275 T^{8} + 700523495472 p^{4} T^{10} + 117868746 p^{8} T^{12} + 14364 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 31512 T^{2} + 469696188 T^{4} - 4285342561320 T^{6} + 26126405445829766 T^{8} - 4285342561320 p^{4} T^{10} + 469696188 p^{8} T^{12} - 31512 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 11064 T^{2} + 131124508 T^{4} - 901551032712 T^{6} + 5798260918327494 T^{8} - 901551032712 p^{4} T^{10} + 131124508 p^{8} T^{12} - 11064 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 35400 T^{2} + 591951516 T^{4} - 6232687244280 T^{6} + 45861936410406470 T^{8} - 6232687244280 p^{4} T^{10} + 591951516 p^{8} T^{12} - 35400 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 83 | \( ( 1 + 6424 T^{2} + 64588572 T^{4} + 770632625960 T^{6} + 4303554072892550 T^{8} + 770632625960 p^{4} T^{10} + 64588572 p^{8} T^{12} + 6424 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 64 T + 28356 T^{2} - 1283264 T^{3} + 322223942 T^{4} - 1283264 p^{2} T^{5} + 28356 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 97 | \( ( 1 - 41636 T^{2} + 941790058 T^{4} - 14310982927568 T^{6} + 156484383916080883 T^{8} - 14310982927568 p^{4} T^{10} + 941790058 p^{8} T^{12} - 41636 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
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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
−2.29783372241479068918878205409, −2.16398443661321986896796673085, −2.13022428386586771735823931556, −2.06917822841894713049346521146, −2.06163413897362031791847746559, −2.00720426535947201796442250677, −2.00547218826817322558981284064, −1.83387927749610307783024575642, −1.74844549468098156533544457029, −1.68433979146283712418535939617, −1.67787852379047628641609525289, −1.51515319077821033863677055098, −1.50089882768097612901954102052, −1.32867838817874305884086653546, −1.14438824681343145317772979501, −1.00273781365372283340967135486, −0.909052258909545388650335526667, −0.891157833263681942520345168106, −0.77797889258918072502747193559, −0.76316794231924306580202958544, −0.70766537275434384426676424088, −0.43652926094087877008061153399, −0.24291067041490851694938003733, −0.18781047299011237784988792107, −0.01901633164696828676872959186,
0.01901633164696828676872959186, 0.18781047299011237784988792107, 0.24291067041490851694938003733, 0.43652926094087877008061153399, 0.70766537275434384426676424088, 0.76316794231924306580202958544, 0.77797889258918072502747193559, 0.891157833263681942520345168106, 0.909052258909545388650335526667, 1.00273781365372283340967135486, 1.14438824681343145317772979501, 1.32867838817874305884086653546, 1.50089882768097612901954102052, 1.51515319077821033863677055098, 1.67787852379047628641609525289, 1.68433979146283712418535939617, 1.74844549468098156533544457029, 1.83387927749610307783024575642, 2.00547218826817322558981284064, 2.00720426535947201796442250677, 2.06163413897362031791847746559, 2.06917822841894713049346521146, 2.13022428386586771735823931556, 2.16398443661321986896796673085, 2.29783372241479068918878205409
Plot not available for L-functions of degree greater than 10.
