| L(s) = 1 | − 4-s − 3·16-s + 8·19-s − 40·49-s − 5·64-s + 48·71-s − 8·76-s + 4·81-s + 16·101-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 40·196-s + 197-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3/4·16-s + 1.83·19-s − 5.71·49-s − 5/8·64-s + 5.69·71-s − 0.917·76-s + 4/9·81-s + 1.59·101-s + 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 20/7·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.986835133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986835133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{2} + p^{2} T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 - 4 T^{4} + 16 T^{6} + 70 T^{8} + 16 p^{2} T^{10} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 20 T^{2} + 244 T^{4} + 2364 T^{6} + 18678 T^{8} + 2364 p^{2} T^{10} + 244 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 14424 p^{2} T^{10} + 892 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 4 p T^{2} + 1556 T^{4} + 31660 T^{6} + 477814 T^{8} + 31660 p^{2} T^{10} + 1556 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 84 T^{2} + 212 p T^{4} + 101164 T^{6} + 2018902 T^{8} + 101164 p^{2} T^{10} + 212 p^{5} T^{12} + 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 2 T + 40 T^{2} - 42 T^{3} + 766 T^{4} - 42 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 - 4 p T^{2} + 4420 T^{4} - 144852 T^{6} + 3702742 T^{8} - 144852 p^{2} T^{10} + 4420 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 52 T^{2} - 112 T^{3} + 1286 T^{4} - 112 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 170900 p^{2} T^{10} + 5412 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 68 T^{2} + 4788 T^{4} + 250460 T^{6} + 9311478 T^{8} + 250460 p^{2} T^{10} + 4788 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 2456248 p^{2} T^{10} + 32668 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 320 T^{2} + 45756 T^{4} - 3816624 T^{6} + 203071110 T^{8} - 3816624 p^{2} T^{10} + 45756 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 140 T^{2} + 284 p T^{4} - 854628 T^{6} + 45549910 T^{8} - 854628 p^{2} T^{10} + 284 p^{5} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 312 T^{2} + 45948 T^{4} - 4212040 T^{6} + 265479398 T^{8} - 4212040 p^{2} T^{10} + 45948 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 3304648 p^{2} T^{10} + 41500 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 3783016 p^{2} T^{10} + 39452 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 176 T^{2} + 15964 T^{4} - 993216 T^{6} + 63719302 T^{8} - 993216 p^{2} T^{10} + 15964 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 12 T + 220 T^{2} - 1564 T^{3} + 18854 T^{4} - 1564 p T^{5} + 220 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 344 T^{2} + 60316 T^{4} - 7095528 T^{6} + 604376710 T^{8} - 7095528 p^{2} T^{10} + 60316 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 18135668 p^{2} T^{10} + 125604 p^{4} T^{12} - 524 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 512 T^{2} + 125148 T^{4} + 18790160 T^{6} + 1886502918 T^{8} + 18790160 p^{2} T^{10} + 125148 p^{4} T^{12} + 512 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 8336760 p^{2} T^{10} + 61788 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 408 T^{2} + 72412 T^{4} - 7810088 T^{6} + 720805318 T^{8} - 7810088 p^{2} T^{10} + 72412 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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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
−2.81971437793366528604785632520, −2.73467940180429537949133914303, −2.61884221564518203322847700296, −2.59571861270832167002548658286, −2.55895087809225747821236549116, −2.48335847960520698040051565479, −2.36203655454983665483009812218, −2.23812153398950446355758649975, −2.17920329203155986654847694258, −2.06561117371089591583693374102, −2.03569870843319648088644928773, −1.98247445607578083518216549422, −1.86809360848368401651626968453, −1.64447049002211886703075764421, −1.51865017370737165320144446033, −1.50044227595646367225993720138, −1.29474823675554038755688075771, −1.23300653408837132333477135808, −1.10248650905884701525080644304, −1.07823822080919812320994713343, −0.950393878809377982746422250319, −0.810269246833016453136966247583, −0.38714987123489692242994129132, −0.29773169691891100667852347425, −0.18540999638234605428406396590,
0.18540999638234605428406396590, 0.29773169691891100667852347425, 0.38714987123489692242994129132, 0.810269246833016453136966247583, 0.950393878809377982746422250319, 1.07823822080919812320994713343, 1.10248650905884701525080644304, 1.23300653408837132333477135808, 1.29474823675554038755688075771, 1.50044227595646367225993720138, 1.51865017370737165320144446033, 1.64447049002211886703075764421, 1.86809360848368401651626968453, 1.98247445607578083518216549422, 2.03569870843319648088644928773, 2.06561117371089591583693374102, 2.17920329203155986654847694258, 2.23812153398950446355758649975, 2.36203655454983665483009812218, 2.48335847960520698040051565479, 2.55895087809225747821236549116, 2.59571861270832167002548658286, 2.61884221564518203322847700296, 2.73467940180429537949133914303, 2.81971437793366528604785632520
Plot not available for L-functions of degree greater than 10.
