| L(s) = 1 | + 4·3-s + 2·7-s + 6·9-s − 4·11-s + 4·13-s − 2·17-s − 4·19-s + 8·21-s + 6·23-s − 12·29-s − 16·33-s + 4·37-s + 16·39-s + 24·41-s − 32·43-s + 2·47-s − 8·51-s + 20·53-s − 16·57-s + 14·59-s − 16·61-s + 12·63-s + 18·67-s + 24·69-s − 28·71-s − 8·73-s − 8·77-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.755·7-s + 2·9-s − 1.20·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s + 1.74·21-s + 1.25·23-s − 2.22·29-s − 2.78·33-s + 0.657·37-s + 2.56·39-s + 3.74·41-s − 4.87·43-s + 0.291·47-s − 1.12·51-s + 2.74·53-s − 2.11·57-s + 1.82·59-s − 2.04·61-s + 1.51·63-s + 2.19·67-s + 2.88·69-s − 3.32·71-s − 0.936·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.33813434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.33813434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2 T + 4 T^{2} + 22 T^{3} - 83 T^{4} + 22 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 11 | \( 1 + 4 T - 2 p T^{2} - 68 T^{3} + 36 p T^{4} + 526 T^{5} - 6648 T^{6} - 124 p T^{7} + 89791 T^{8} - 124 p^{2} T^{9} - 6648 p^{2} T^{10} + 526 p^{3} T^{11} + 36 p^{5} T^{12} - 68 p^{5} T^{13} - 2 p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 2 T + 28 T^{2} - 6 p T^{3} + 465 T^{4} - 6 p^{2} T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 2 T - 40 T^{2} - 144 T^{3} + 788 T^{4} + 3412 T^{5} - 5360 T^{6} - 32022 T^{7} + 6583 T^{8} - 32022 p T^{9} - 5360 p^{2} T^{10} + 3412 p^{3} T^{11} + 788 p^{4} T^{12} - 144 p^{5} T^{13} - 40 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 4 T - 18 T^{2} + 112 T^{3} + 697 T^{4} - 2652 T^{5} + 9806 T^{6} + 65960 T^{7} - 161964 T^{8} + 65960 p T^{9} + 9806 p^{2} T^{10} - 2652 p^{3} T^{11} + 697 p^{4} T^{12} + 112 p^{5} T^{13} - 18 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 6 T - 44 T^{2} + 272 T^{3} + 1468 T^{4} - 6616 T^{5} - 39952 T^{6} + 50582 T^{7} + 1119911 T^{8} + 50582 p T^{9} - 39952 p^{2} T^{10} - 6616 p^{3} T^{11} + 1468 p^{4} T^{12} + 272 p^{5} T^{13} - 44 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 6 T + 50 T^{2} + 300 T^{3} + 2316 T^{4} + 300 p T^{5} + 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 22 T^{2} + 400 T^{3} - 59 T^{4} - 10600 T^{5} + 70338 T^{6} + 164200 T^{7} - 2333796 T^{8} + 164200 p T^{9} + 70338 p^{2} T^{10} - 10600 p^{3} T^{11} - 59 p^{4} T^{12} + 400 p^{5} T^{13} - 22 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 4 T - 108 T^{2} + 292 T^{3} + 7187 T^{4} - 10530 T^{5} - 383576 T^{6} + 128186 T^{7} + 16685028 T^{8} + 128186 p T^{9} - 383576 p^{2} T^{10} - 10530 p^{3} T^{11} + 7187 p^{4} T^{12} + 292 p^{5} T^{13} - 108 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 12 T + 206 T^{2} - 1514 T^{3} + 13536 T^{4} - 1514 p T^{5} + 206 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 16 T + 160 T^{2} + 1402 T^{3} + 10561 T^{4} + 1402 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 2 T - 64 T^{2} - 84 T^{3} + 1982 T^{4} + 7370 T^{5} + 122536 T^{6} - 409218 T^{7} - 7587185 T^{8} - 409218 p T^{9} + 122536 p^{2} T^{10} + 7370 p^{3} T^{11} + 1982 p^{4} T^{12} - 84 p^{5} T^{13} - 64 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 20 T + 68 T^{2} + 200 T^{3} + 12154 T^{4} - 129900 T^{5} + 85936 T^{6} - 2744060 T^{7} + 61991603 T^{8} - 2744060 p T^{9} + 85936 p^{2} T^{10} - 129900 p^{3} T^{11} + 12154 p^{4} T^{12} + 200 p^{5} T^{13} + 68 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 - 14 T + 98 T^{2} - 2364 T^{3} + 23888 T^{4} - 127648 T^{5} + 2057308 T^{6} - 17027622 T^{7} + 74320339 T^{8} - 17027622 p T^{9} + 2057308 p^{2} T^{10} - 127648 p^{3} T^{11} + 23888 p^{4} T^{12} - 2364 p^{5} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 16 T + 48 T^{2} - 408 T^{3} - 42 p T^{4} - 7524 T^{5} - 100912 T^{6} - 1615000 T^{7} - 17451933 T^{8} - 1615000 p T^{9} - 100912 p^{2} T^{10} - 7524 p^{3} T^{11} - 42 p^{5} T^{12} - 408 p^{5} T^{13} + 48 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 18 T + 20 T^{2} + 1396 T^{3} - 1157 T^{4} - 130618 T^{5} + 724920 T^{6} + 842608 T^{7} - 21438852 T^{8} + 842608 p T^{9} + 724920 p^{2} T^{10} - 130618 p^{3} T^{11} - 1157 p^{4} T^{12} + 1396 p^{5} T^{13} + 20 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 14 T + 158 T^{2} + 444 T^{3} + 3552 T^{4} + 444 p T^{5} + 158 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 8 T - 144 T^{2} - 1148 T^{3} + 10427 T^{4} + 49662 T^{5} - 1117916 T^{6} - 363682 T^{7} + 112617972 T^{8} - 363682 p T^{9} - 1117916 p^{2} T^{10} + 49662 p^{3} T^{11} + 10427 p^{4} T^{12} - 1148 p^{5} T^{13} - 144 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 8 T - 18 T^{2} - 8 p T^{3} + 6121 T^{4} + 6180 T^{5} + 974510 T^{6} - 7889020 T^{7} - 17559900 T^{8} - 7889020 p T^{9} + 974510 p^{2} T^{10} + 6180 p^{3} T^{11} + 6121 p^{4} T^{12} - 8 p^{6} T^{13} - 18 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 10 T + 200 T^{2} - 1808 T^{3} + 24704 T^{4} - 1808 p T^{5} + 200 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 8 T + 2 T^{2} - 1724 T^{3} + 19584 T^{4} - 80534 T^{5} + 2467704 T^{6} - 22993064 T^{7} + 80738407 T^{8} - 22993064 p T^{9} + 2467704 p^{2} T^{10} - 80534 p^{3} T^{11} + 19584 p^{4} T^{12} - 1724 p^{5} T^{13} + 2 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 10 T + 340 T^{2} - 2590 T^{3} + 47986 T^{4} - 2590 p T^{5} + 340 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.85168466201273234208647334364, −3.82663373879623758652757414839, −3.43506345597912413103806549340, −3.42324033638510389899504421025, −3.35487166493550116204749592962, −3.08486034376320731918104663669, −3.05487976498227197600370591185, −3.04598832641435671271983412744, −2.78462505403999467660901722494, −2.65997177230766060160956310451, −2.54185762943627285612190100854, −2.43550434044764528841135381623, −2.41926657378925666118805656997, −2.23306869620293082630387594904, −1.84543589739213898264942752200, −1.82594936113000880465119382631, −1.81243857822671615013752001853, −1.73027763609038921722944078906, −1.68577839241216972022109728152, −1.29108667765670878216908815685, −1.05187507738958953656558115618, −0.78067188420332712143146035688, −0.66556863819920561159234101040, −0.53786072078723185442546860496, −0.23030290966354307770919376996,
0.23030290966354307770919376996, 0.53786072078723185442546860496, 0.66556863819920561159234101040, 0.78067188420332712143146035688, 1.05187507738958953656558115618, 1.29108667765670878216908815685, 1.68577839241216972022109728152, 1.73027763609038921722944078906, 1.81243857822671615013752001853, 1.82594936113000880465119382631, 1.84543589739213898264942752200, 2.23306869620293082630387594904, 2.41926657378925666118805656997, 2.43550434044764528841135381623, 2.54185762943627285612190100854, 2.65997177230766060160956310451, 2.78462505403999467660901722494, 3.04598832641435671271983412744, 3.05487976498227197600370591185, 3.08486034376320731918104663669, 3.35487166493550116204749592962, 3.42324033638510389899504421025, 3.43506345597912413103806549340, 3.82663373879623758652757414839, 3.85168466201273234208647334364
Plot not available for L-functions of degree greater than 10.
