| L(s) = 1 | − 2·2-s + 6·3-s + 4-s + 2·5-s − 12·6-s − 2·7-s + 2·8-s + 18·9-s − 4·10-s + 6·12-s + 4·13-s + 4·14-s + 12·15-s − 4·16-s − 6·17-s − 36·18-s + 4·19-s + 2·20-s − 12·21-s − 8·23-s + 12·24-s + 5·25-s − 8·26-s + 36·27-s − 2·28-s − 8·29-s − 24·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3.46·3-s + 1/2·4-s + 0.894·5-s − 4.89·6-s − 0.755·7-s + 0.707·8-s + 6·9-s − 1.26·10-s + 1.73·12-s + 1.10·13-s + 1.06·14-s + 3.09·15-s − 16-s − 1.45·17-s − 8.48·18-s + 0.917·19-s + 0.447·20-s − 2.61·21-s − 1.66·23-s + 2.44·24-s + 25-s − 1.56·26-s + 6.92·27-s − 0.377·28-s − 1.48·29-s − 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.372248974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.372248974\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T^{2} - 189 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 11 T^{2} - 4 T^{3} + 352 T^{4} - 4 p T^{5} - 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 216 T^{3} + 1211 T^{4} + 216 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 36 T^{3} - 436 T^{4} + 36 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 47 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 104 T^{3} + 82 T^{4} + 104 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 84 T^{3} - 1822 T^{4} - 84 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 68 T^{3} + 2143 T^{4} + 68 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 10 T + 146 T^{2} + 988 T^{3} + 9319 T^{4} + 988 p T^{5} + 146 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 568 T^{3} - 6428 T^{4} - 568 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 22 T + 2 p T^{2} + 1272 T^{3} - 20857 T^{4} + 1272 p T^{5} + 2 p^{3} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 90 T^{2} - 420 T^{3} + 671 T^{4} - 420 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 84 T^{3} - 7954 T^{4} - 84 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 144 T^{2} + 24 p T^{3} - 34273 T^{4} + 24 p^{2} T^{5} + 144 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 + 8 T + 32 T^{2} - 1072 T^{3} - 11177 T^{4} - 1072 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16 T + 185 T^{2} - 556 T^{3} + 2404 T^{4} - 556 p T^{5} + 185 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 38406 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.320236419491236219675546614843, −8.030590904142064870295466686322, −7.958002072989437536038516889344, −7.78566148444406045363332804363, −7.60274799173849203082454050892, −7.21573427275540525187265570235, −6.82877862008383579159926181741, −6.69266371141125155498107924010, −6.54304275085652480987053295365, −5.96965622563691283398462359847, −5.90966451108915705720335090715, −5.38905436381975429039984707839, −5.21731643457830730884678687689, −4.54478443248201076265036428234, −4.49062611548685416506908701067, −3.95410893442601225765652286778, −3.66109746830888744705773493555, −3.58898025443676707729640172775, −3.26037347577718321093170999495, −2.74754758788002533090066621728, −2.59205667456774069942830768555, −2.04184986262065740661930364118, −1.90430715497974095113563083030, −1.64484797522181112494170960862, −0.72997252854848011570873989773,
0.72997252854848011570873989773, 1.64484797522181112494170960862, 1.90430715497974095113563083030, 2.04184986262065740661930364118, 2.59205667456774069942830768555, 2.74754758788002533090066621728, 3.26037347577718321093170999495, 3.58898025443676707729640172775, 3.66109746830888744705773493555, 3.95410893442601225765652286778, 4.49062611548685416506908701067, 4.54478443248201076265036428234, 5.21731643457830730884678687689, 5.38905436381975429039984707839, 5.90966451108915705720335090715, 5.96965622563691283398462359847, 6.54304275085652480987053295365, 6.69266371141125155498107924010, 6.82877862008383579159926181741, 7.21573427275540525187265570235, 7.60274799173849203082454050892, 7.78566148444406045363332804363, 7.958002072989437536038516889344, 8.030590904142064870295466686322, 8.320236419491236219675546614843
