| L(s) = 1 | − 2-s − 2·3-s + 2·4-s − 4·5-s + 2·6-s + 7-s + 3·9-s + 4·10-s − 2·11-s − 4·12-s + 13-s − 14-s + 8·15-s − 17-s − 3·18-s − 8·20-s − 2·21-s + 2·22-s + 2·23-s − 26-s + 2·28-s − 10·29-s − 8·30-s + 4·31-s + 11·32-s + 4·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s + 9-s + 1.26·10-s − 0.603·11-s − 1.15·12-s + 0.277·13-s − 0.267·14-s + 2.06·15-s − 0.242·17-s − 0.707·18-s − 1.78·20-s − 0.436·21-s + 0.426·22-s + 0.417·23-s − 0.196·26-s + 0.377·28-s − 1.85·29-s − 1.46·30-s + 0.718·31-s + 1.94·32-s + 0.696·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7273666458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7273666458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + T - T^{2} - 3 T^{3} - T^{4} - 3 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 4 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2:C_4$ | \( 1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 17 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2:C_4$ | \( 1 + T - T^{2} - 53 T^{3} + 104 T^{4} - 53 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 2 T + T^{2} - 106 T^{3} + 729 T^{4} - 106 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 10 T + 71 T^{2} + 520 T^{3} + 3641 T^{4} + 520 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 18 T + 101 T^{2} - 174 T^{3} + 49 T^{4} - 174 p T^{5} + 101 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 16 T + 59 T^{2} + 738 T^{3} - 9721 T^{4} + 738 p T^{5} + 59 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 2 T - 49 T^{2} + 204 T^{3} + 2189 T^{4} + 204 p T^{5} - 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 15 T + 41 T^{2} - 45 T^{3} + 1156 T^{4} - 45 p T^{5} + 41 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 23 T + 253 T^{2} + 23 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 11 T + 163 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 2 T - 67 T^{2} - 74 T^{3} + 5255 T^{4} - 74 p T^{5} - 67 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 850 p T^{5} + 63 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 24 T + 297 T^{2} - 3382 T^{3} + 35205 T^{4} - 3382 p T^{5} + 297 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - T - 67 T^{2} + 515 T^{3} + 5516 T^{4} + 515 p T^{5} - 67 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 4 T - 73 T^{2} + 648 T^{3} + 3905 T^{4} + 648 p T^{5} - 73 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 28 T + 287 T^{2} - 1970 T^{3} + 17821 T^{4} - 1970 p T^{5} + 287 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.056436236349321620877424597651, −7.68055329408569500775825804039, −7.65997345608286812904613595233, −7.49333102890119888557642701961, −7.38904886825371514209756126784, −7.17483196850584538117256010088, −6.53223801940625182425229158734, −6.52513670524754600640636064737, −6.08740666144220429149862523119, −5.97130090661033623977551299443, −5.83939984082801459415556311334, −5.21769180166285259437882935223, −5.20208205010695879205138925116, −4.59124833446407140663878267951, −4.40653707215554206586527704777, −4.35550831915590512595117216231, −4.03949257526899958612328440128, −3.62073719996397896498502552822, −3.26123151627102795157530440873, −2.77272096739335599497878422339, −2.57109930279020162226778581215, −1.96981087112958611634065820052, −1.62616780210148452730842898166, −0.921693494818302228843595773958, −0.51631928000856545030200929995,
0.51631928000856545030200929995, 0.921693494818302228843595773958, 1.62616780210148452730842898166, 1.96981087112958611634065820052, 2.57109930279020162226778581215, 2.77272096739335599497878422339, 3.26123151627102795157530440873, 3.62073719996397896498502552822, 4.03949257526899958612328440128, 4.35550831915590512595117216231, 4.40653707215554206586527704777, 4.59124833446407140663878267951, 5.20208205010695879205138925116, 5.21769180166285259437882935223, 5.83939984082801459415556311334, 5.97130090661033623977551299443, 6.08740666144220429149862523119, 6.52513670524754600640636064737, 6.53223801940625182425229158734, 7.17483196850584538117256010088, 7.38904886825371514209756126784, 7.49333102890119888557642701961, 7.65997345608286812904613595233, 7.68055329408569500775825804039, 8.056436236349321620877424597651
