| L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s + 4·5-s + 16·6-s + 4·7-s − 20·8-s + 10·9-s − 16·10-s + 9·11-s − 40·12-s + 4·13-s − 16·14-s − 16·15-s + 35·16-s + 4·17-s − 40·18-s − 19-s + 40·20-s − 16·21-s − 36·22-s + 6·23-s + 80·24-s + 25-s − 16·26-s − 20·27-s + 40·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.51·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s + 2.71·11-s − 11.5·12-s + 1.10·13-s − 4.27·14-s − 4.13·15-s + 35/4·16-s + 0.970·17-s − 9.42·18-s − 0.229·19-s + 8.94·20-s − 3.49·21-s − 7.67·22-s + 1.25·23-s + 16.3·24-s + 1/5·25-s − 3.13·26-s − 3.84·27-s + 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.604475825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.604475825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 4 T + 3 p T^{2} - 7 p T^{3} + 82 T^{4} - 7 p^{2} T^{5} + 3 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 9 T + 59 T^{2} - 272 T^{3} + 1048 T^{4} - 272 p T^{5} + 59 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + T + 54 T^{2} + 94 T^{3} + 1314 T^{4} + 94 p T^{5} + 54 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 6 T + 73 T^{2} - 337 T^{3} + 2252 T^{4} - 337 p T^{5} + 73 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 5 T + 86 T^{2} + 360 T^{3} + 3556 T^{4} + 360 p T^{5} + 86 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 9 T + 88 T^{2} - 472 T^{3} + 3454 T^{4} - 472 p T^{5} + 88 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + T + 82 T^{2} - 34 T^{3} + 3780 T^{4} - 34 p T^{5} + 82 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 13 T + 196 T^{2} - 1548 T^{3} + 12664 T^{4} - 1548 p T^{5} + 196 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 7 T + 118 T^{2} - 518 T^{3} + 6150 T^{4} - 518 p T^{5} + 118 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 11 T + 220 T^{2} - 1556 T^{3} + 16222 T^{4} - 1556 p T^{5} + 220 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 9 T + 98 T^{2} - 818 T^{3} + 9132 T^{4} - 818 p T^{5} + 98 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 203 T^{2} + 55 T^{3} + 17052 T^{4} + 55 p T^{5} + 203 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + T + 134 T^{2} + 124 T^{3} + 9772 T^{4} + 124 p T^{5} + 134 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 9 T + 180 T^{2} + 1874 T^{3} + 15182 T^{4} + 1874 p T^{5} + 180 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 17 T + 4 p T^{2} - 2944 T^{3} + 30522 T^{4} - 2944 p T^{5} + 4 p^{3} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 20 T + 345 T^{2} - 3435 T^{3} + 34958 T^{4} - 3435 p T^{5} + 345 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 5 T + 210 T^{2} + 1278 T^{3} + 21042 T^{4} + 1278 p T^{5} + 210 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 5 T + 282 T^{2} - 1324 T^{3} + 33022 T^{4} - 1324 p T^{5} + 282 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 11 T + 360 T^{2} + 2662 T^{3} + 47512 T^{4} + 2662 p T^{5} + 360 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - T + 222 T^{2} - 1036 T^{3} + 23784 T^{4} - 1036 p T^{5} + 222 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.69651904898458977405875807745, −5.21361010274128069009928486288, −5.16487945650249794120769754844, −5.13346664177890739203716783798, −4.87624399841039151431422901989, −4.46636108661590863360093636825, −4.28184801664236322490993275729, −4.14071598018052770114776471811, −4.03196234678341597973352045206, −3.73536989636071694651978765237, −3.53285843016535977970215225769, −3.35587482129725843807366742875, −3.10683862858260349240437070504, −2.44251149904359164513123545125, −2.43130769931193248276123204114, −2.34016545571479602167103236627, −2.24973487778575300048984787606, −1.60445805549699366919021740273, −1.55114859283068617014771088544, −1.45690670357718104295367126018, −1.41774245869009570331909541389, −0.942603335637177770356638841914, −0.78338560925844350013861710418, −0.64385548215945005423895763582, −0.56868474979881480836854760643,
0.56868474979881480836854760643, 0.64385548215945005423895763582, 0.78338560925844350013861710418, 0.942603335637177770356638841914, 1.41774245869009570331909541389, 1.45690670357718104295367126018, 1.55114859283068617014771088544, 1.60445805549699366919021740273, 2.24973487778575300048984787606, 2.34016545571479602167103236627, 2.43130769931193248276123204114, 2.44251149904359164513123545125, 3.10683862858260349240437070504, 3.35587482129725843807366742875, 3.53285843016535977970215225769, 3.73536989636071694651978765237, 4.03196234678341597973352045206, 4.14071598018052770114776471811, 4.28184801664236322490993275729, 4.46636108661590863360093636825, 4.87624399841039151431422901989, 5.13346664177890739203716783798, 5.16487945650249794120769754844, 5.21361010274128069009928486288, 5.69651904898458977405875807745
