| L(s) = 1 | − 8·4-s − 8·7-s + 36·16-s − 32·25-s + 64·28-s − 16·37-s + 48·43-s + 36·49-s − 120·64-s + 16·67-s − 16·79-s + 256·100-s − 96·109-s − 288·112-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 208·169-s − 384·172-s + ⋯ |
| L(s) = 1 | − 4·4-s − 3.02·7-s + 9·16-s − 6.39·25-s + 12.0·28-s − 2.63·37-s + 7.31·43-s + 36/7·49-s − 15·64-s + 1.95·67-s − 1.80·79-s + 25.5·100-s − 9.19·109-s − 27.2·112-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 16·169-s − 29.2·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146716094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146716094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + 122 T^{4} + 4 p^{2} T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + T^{2} )^{8} \) |
| good | 5 | \( ( 1 + 16 T^{2} + 98 T^{4} + 192 T^{6} - 158 T^{8} + 192 p^{2} T^{10} + 98 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - p T^{2} )^{16} \) |
| 17 | \( ( 1 + 52 T^{2} + 1910 T^{4} + 47364 T^{6} + 54914 p T^{8} + 47364 p^{2} T^{10} + 1910 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 48 T^{2} + 962 T^{4} - 25584 T^{6} + 675522 T^{8} - 25584 p^{2} T^{10} + 962 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 64 T^{2} + 2876 T^{4} - 85440 T^{6} + 2216902 T^{8} - 85440 p^{2} T^{10} + 2876 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 124 T^{2} + 7688 T^{4} - 335700 T^{6} + 11178478 T^{8} - 335700 p^{2} T^{10} + 7688 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 + 20 T^{2} + 518 T^{4} + 13140 T^{6} + 1290050 T^{8} + 13140 p^{2} T^{10} + 518 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 4 T + 96 T^{2} + 316 T^{3} + 4910 T^{4} + 316 p T^{5} + 96 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 196 T^{2} + 17750 T^{4} + 1029876 T^{6} + 46163170 T^{8} + 1029876 p^{2} T^{10} + 17750 p^{4} T^{12} + 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 12 T + 198 T^{2} - 1484 T^{3} + 13314 T^{4} - 1484 p T^{5} + 198 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 180 T^{2} + 14054 T^{4} + 721620 T^{6} + 33358914 T^{8} + 721620 p^{2} T^{10} + 14054 p^{4} T^{12} + 180 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 164 T^{2} + 18488 T^{4} - 1378668 T^{6} + 84038222 T^{8} - 1378668 p^{2} T^{10} + 18488 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 376 T^{2} + 1108 p T^{4} + 6923400 T^{6} + 492070054 T^{8} + 6923400 p^{2} T^{10} + 1108 p^{5} T^{12} + 376 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 40 T^{2} + 92 T^{4} - 287400 T^{6} - 11177114 T^{8} - 287400 p^{2} T^{10} + 92 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 4 T + 198 T^{2} - 628 T^{3} + 17570 T^{4} - 628 p T^{5} + 198 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 256 T^{2} + 33788 T^{4} - 3475200 T^{6} + 285337798 T^{8} - 3475200 p^{2} T^{10} + 33788 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 388 T^{2} + 70838 T^{4} - 8296980 T^{6} + 701448674 T^{8} - 8296980 p^{2} T^{10} + 70838 p^{4} T^{12} - 388 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 4 T + 140 T^{2} + 108 T^{3} + 9346 T^{4} + 108 p T^{5} + 140 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 316 T^{2} + 52870 T^{4} + 6559612 T^{6} + 628398466 T^{8} + 6559612 p^{2} T^{10} + 52870 p^{4} T^{12} + 316 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 292 T^{2} + 38216 T^{4} + 3178860 T^{6} + 248573134 T^{8} + 3178860 p^{2} T^{10} + 38216 p^{4} T^{12} + 292 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 296 T^{2} + 53852 T^{4} - 7384344 T^{6} + 801294022 T^{8} - 7384344 p^{2} T^{10} + 53852 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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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
−2.48558810479706413478944954104, −2.48072636785076359739177355217, −2.31524535992871579662875549010, −2.25660079603352846919244387077, −2.12297871926807049308506228298, −2.01019586586484744710829241741, −2.00412784936308371621538383241, −1.97429731284927676804491253368, −1.75314653620608062226788221596, −1.63989062515641759359024506570, −1.60222225534461171303534266997, −1.56256071293657096943907100737, −1.44619804943248609932544687152, −1.33006595297265707933620384409, −1.25383369375988815557017479881, −1.09413389785414146625048930157, −1.02024913695357798336156823353, −1.01976969820193861913893217472, −0.54415978433004949756262414996, −0.53318983015303867568217978450, −0.52714240050751429611538598692, −0.48279924590825292288324225344, −0.42641383156806202489021768208, −0.23962020225016066391790956021, −0.20710212772328822983037646612,
0.20710212772328822983037646612, 0.23962020225016066391790956021, 0.42641383156806202489021768208, 0.48279924590825292288324225344, 0.52714240050751429611538598692, 0.53318983015303867568217978450, 0.54415978433004949756262414996, 1.01976969820193861913893217472, 1.02024913695357798336156823353, 1.09413389785414146625048930157, 1.25383369375988815557017479881, 1.33006595297265707933620384409, 1.44619804943248609932544687152, 1.56256071293657096943907100737, 1.60222225534461171303534266997, 1.63989062515641759359024506570, 1.75314653620608062226788221596, 1.97429731284927676804491253368, 2.00412784936308371621538383241, 2.01019586586484744710829241741, 2.12297871926807049308506228298, 2.25660079603352846919244387077, 2.31524535992871579662875549010, 2.48072636785076359739177355217, 2.48558810479706413478944954104
Plot not available for L-functions of degree greater than 10.
