| L(s) = 1 | + 4·2-s + 8·4-s + 12·7-s + 8·8-s + 48·14-s + 4·16-s + 96·28-s − 16·31-s + 8·32-s − 24·41-s + 24·47-s + 72·49-s + 96·56-s − 64·62-s + 32·64-s − 60·67-s + 72·71-s + 48·81-s − 96·82-s + 96·94-s + 72·97-s + 288·98-s + 64·101-s − 8·103-s − 48·107-s + 48·112-s + 60·113-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 4.53·7-s + 2.82·8-s + 12.8·14-s + 16-s + 18.1·28-s − 2.87·31-s + 1.41·32-s − 3.74·41-s + 3.50·47-s + 72/7·49-s + 12.8·56-s − 8.12·62-s + 4·64-s − 7.33·67-s + 8.54·71-s + 16/3·81-s − 10.6·82-s + 9.90·94-s + 7.31·97-s + 29.0·98-s + 6.36·101-s − 0.788·103-s − 4.64·107-s + 4.53·112-s + 5.64·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(582.3621643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(582.3621643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( ( 1 + 8 T + 60 T^{2} + 376 T^{3} + 1398 T^{4} + 376 p T^{5} + 60 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| good | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{4} + p^{2} T^{5} + p^{2} T^{8} + p^{5} T^{11} - p^{6} T^{12} + p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2} \) |
| 3 | \( ( 1 - 2 p T + 2 p^{2} T^{2} - 14 p T^{3} + 22 p T^{4} - 8 p T^{5} - 2 p^{4} T^{6} + 68 p^{2} T^{7} - 1349 T^{8} + 68 p^{3} T^{9} - 2 p^{6} T^{10} - 8 p^{4} T^{11} + 22 p^{5} T^{12} - 14 p^{6} T^{13} + 2 p^{8} T^{14} - 2 p^{8} T^{15} + p^{8} T^{16} )( 1 + 2 p T + 2 p^{2} T^{2} + 14 p T^{3} + 22 p T^{4} + 8 p T^{5} - 2 p^{4} T^{6} - 68 p^{2} T^{7} - 1349 T^{8} - 68 p^{3} T^{9} - 2 p^{6} T^{10} + 8 p^{4} T^{11} + 22 p^{5} T^{12} + 14 p^{6} T^{13} + 2 p^{8} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} ) \) |
| 7 | \( ( 1 - 6 T + 18 T^{2} - 62 T^{3} + 260 T^{4} - 794 T^{5} + 2006 T^{6} - 6218 T^{7} + 19002 T^{8} - 6218 p T^{9} + 2006 p^{2} T^{10} - 794 p^{3} T^{11} + 260 p^{4} T^{12} - 62 p^{5} T^{13} + 18 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 116 T^{4} + 344 T^{6} - 6686 T^{8} + 344 p^{2} T^{10} + 116 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( 1 - 196 T^{4} + 50828 T^{8} - 12133980 T^{12} + 1459609206 T^{16} - 12133980 p^{4} T^{20} + 50828 p^{8} T^{24} - 196 p^{12} T^{28} + p^{16} T^{32} \) |
| 17 | \( 1 - 124 T^{4} + 94754 T^{8} - 38928012 T^{12} + 8832888627 T^{16} - 38928012 p^{4} T^{20} + 94754 p^{8} T^{24} - 124 p^{12} T^{28} + p^{16} T^{32} \) |
| 19 | \( ( 1 - 124 T^{2} + 7130 T^{4} - 248460 T^{6} + 5734059 T^{8} - 248460 p^{2} T^{10} + 7130 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( 1 - 84 T^{4} + 321068 T^{8} + 270123380 T^{12} - 11101222410 T^{16} + 270123380 p^{4} T^{20} + 321068 p^{8} T^{24} - 84 p^{12} T^{28} + p^{16} T^{32} \) |
| 29 | \( ( 1 + 80 T^{2} + 4424 T^{4} + 160516 T^{6} + 5200762 T^{8} + 160516 p^{2} T^{10} + 4424 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( 1 + 604 T^{4} - 3848190 T^{8} - 322626700 T^{12} + 9858346598675 T^{16} - 322626700 p^{4} T^{20} - 3848190 p^{8} T^{24} + 604 p^{12} T^{28} + p^{16} T^{32} \) |
| 41 | \( ( 1 + 6 T + 120 T^{2} + 420 T^{3} + 6009 T^{4} + 420 p T^{5} + 120 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 43 | \( 1 + 204 T^{4} - 1920862 T^{8} - 2304605500 T^{12} + 7574617760451 T^{16} - 2304605500 p^{4} T^{20} - 1920862 p^{8} T^{24} + 204 p^{12} T^{28} + p^{16} T^{32} \) |
| 47 | \( ( 1 - 12 T + 72 T^{2} - 724 T^{3} + 10468 T^{4} - 74884 T^{5} + 407000 T^{6} - 3802844 T^{7} + 35310150 T^{8} - 3802844 p T^{9} + 407000 p^{2} T^{10} - 74884 p^{3} T^{11} + 10468 p^{4} T^{12} - 724 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 53 | \( 1 - 2472 T^{4} + 17235362 T^{8} - 68301188484 T^{12} + 146883981835171 T^{16} - 68301188484 p^{4} T^{20} + 17235362 p^{8} T^{24} - 2472 p^{12} T^{28} + p^{16} T^{32} \) |
| 59 | \( ( 1 - 348 T^{2} + 57794 T^{4} - 5980752 T^{6} + 422102539 T^{8} - 5980752 p^{2} T^{10} + 57794 p^{4} T^{12} - 348 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 184 T^{2} + 19380 T^{4} - 1493240 T^{6} + 93806630 T^{8} - 1493240 p^{2} T^{10} + 19380 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 + 30 T + 450 T^{2} + 5002 T^{3} + 36840 T^{4} + 85878 T^{5} - 1491658 T^{6} - 28478302 T^{7} - 298563566 T^{8} - 28478302 p T^{9} - 1491658 p^{2} T^{10} + 85878 p^{3} T^{11} + 36840 p^{4} T^{12} + 5002 p^{5} T^{13} + 450 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 18 T + 342 T^{2} - 3574 T^{3} + 37931 T^{4} - 3574 p T^{5} + 342 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( 1 - 4200 T^{4} + 49880786 T^{8} - 10647791268 T^{12} + 876465675848659 T^{16} - 10647791268 p^{4} T^{20} + 49880786 p^{8} T^{24} - 4200 p^{12} T^{28} + p^{16} T^{32} \) |
| 79 | \( ( 1 + 308 T^{2} + 50364 T^{4} + 5691256 T^{6} + 500588498 T^{8} + 5691256 p^{2} T^{10} + 50364 p^{4} T^{12} + 308 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( 1 + 1732 T^{4} + 48034434 T^{8} + 123787400516 T^{12} + 217137216949667 T^{16} + 123787400516 p^{4} T^{20} + 48034434 p^{8} T^{24} + 1732 p^{12} T^{28} + p^{16} T^{32} \) |
| 89 | \( ( 1 + 204 T^{2} + 15028 T^{4} + 1461440 T^{6} + 184210610 T^{8} + 1461440 p^{2} T^{10} + 15028 p^{4} T^{12} + 204 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 36 T + 648 T^{2} - 9860 T^{3} + 149796 T^{4} - 1963788 T^{5} + 22238360 T^{6} - 244599148 T^{7} + 2551946566 T^{8} - 244599148 p T^{9} + 22238360 p^{2} T^{10} - 1963788 p^{3} T^{11} + 149796 p^{4} T^{12} - 9860 p^{5} T^{13} + 648 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
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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
−2.66239161322825693928364847607, −2.65567450527127361221349752633, −2.64718216696052170010239602105, −2.56358679497570834169809279742, −2.55485181452538862675242523796, −2.20488154805827431653471632568, −2.17652174636095947131712701599, −2.03541970034583652730788571979, −2.02589823135090500227011786116, −1.97249686512620783376651432547, −1.82953437591191579172077070373, −1.82379976516770773288537143780, −1.78948703468508908968358275171, −1.78947777798892599928802488631, −1.69743964584806559481009623014, −1.53059376630251080874173378053, −1.44990754928502192013821376145, −1.38020916584847995044576745834, −0.946982278382232162687867447922, −0.794482594125963411140355124949, −0.793636211133994949662726822293, −0.77189709257232157371051524116, −0.58047551816979940768287340068, −0.55018195508945231827894783702, −0.49608662284933345029696100431,
0.49608662284933345029696100431, 0.55018195508945231827894783702, 0.58047551816979940768287340068, 0.77189709257232157371051524116, 0.793636211133994949662726822293, 0.794482594125963411140355124949, 0.946982278382232162687867447922, 1.38020916584847995044576745834, 1.44990754928502192013821376145, 1.53059376630251080874173378053, 1.69743964584806559481009623014, 1.78947777798892599928802488631, 1.78948703468508908968358275171, 1.82379976516770773288537143780, 1.82953437591191579172077070373, 1.97249686512620783376651432547, 2.02589823135090500227011786116, 2.03541970034583652730788571979, 2.17652174636095947131712701599, 2.20488154805827431653471632568, 2.55485181452538862675242523796, 2.56358679497570834169809279742, 2.64718216696052170010239602105, 2.65567450527127361221349752633, 2.66239161322825693928364847607
Plot not available for L-functions of degree greater than 10.
