| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 6·5-s + 2·6-s − 7·7-s − 5·8-s − 2·9-s + 12·10-s + 5·11-s − 2·12-s + 13-s + 14·14-s + 6·15-s + 5·16-s − 15·17-s + 4·18-s + 2·19-s − 12·20-s + 7·21-s − 10·22-s − 8·23-s + 5·24-s + 5·25-s − 2·26-s − 14·28-s − 8·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 2.68·5-s + 0.816·6-s − 2.64·7-s − 1.76·8-s − 2/3·9-s + 3.79·10-s + 1.50·11-s − 0.577·12-s + 0.277·13-s + 3.74·14-s + 1.54·15-s + 5/4·16-s − 3.63·17-s + 0.942·18-s + 0.458·19-s − 2.68·20-s + 1.52·21-s − 2.13·22-s − 1.66·23-s + 1.02·24-s + 25-s − 0.392·26-s − 2.64·28-s − 1.48·29-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2:C_4$ | \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + T + p T^{2} + 5 T^{3} + 16 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2:C_4$ | \( 1 + p T + 27 T^{2} + 95 T^{3} + 296 T^{4} + 95 p T^{5} + 27 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 5 T + 4 T^{2} - 25 T^{3} + 201 T^{4} - 25 p T^{5} + 4 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 15 T + 118 T^{2} + 675 T^{3} + 3079 T^{4} + 675 p T^{5} + 118 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 2 T - 15 T^{2} + 68 T^{3} + 149 T^{4} + 68 p T^{5} - 15 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 8 T + T^{2} - 146 T^{3} - 651 T^{4} - 146 p T^{5} + p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 8 T + 5 T^{2} - 8 p T^{3} - 59 p T^{4} - 8 p^{2} T^{5} + 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 13 T + 105 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 19 T + 95 T^{2} - 599 T^{3} - 8776 T^{4} - 599 p T^{5} + 95 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 6 T + 93 T^{2} - 350 T^{3} + 4851 T^{4} - 350 p T^{5} + 93 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 11 T + 49 T^{2} - 547 T^{3} + 5964 T^{4} - 547 p T^{5} + 49 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 6 T + 83 T^{2} + 480 T^{3} + 6481 T^{4} + 480 p T^{5} + 83 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 14 T + 37 T^{2} + 142 T^{3} + 3555 T^{4} + 142 p T^{5} + 37 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 133 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 34 T + 665 T^{2} - 8926 T^{3} + 86729 T^{4} - 8926 p T^{5} + 665 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 5 T + 12 T^{2} - 655 T^{3} + 8639 T^{4} - 655 p T^{5} + 12 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 21 T + 162 T^{2} + 17 p T^{3} + 195 p T^{4} + 17 p^{2} T^{5} + 162 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 12 T + 61 T^{2} + 1146 T^{3} + 17149 T^{4} + 1146 p T^{5} + 61 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + T - 38 T^{2} - 727 T^{3} + 5055 T^{4} - 727 p T^{5} - 38 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 8 T + 87 T^{2} + 1270 T^{3} + 19901 T^{4} + 1270 p T^{5} + 87 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.714251004821748918159059418163, −8.561359501563538684735142171081, −8.421247875333838184784577305652, −7.88174866995798798603224743548, −7.76975535155222827325160735093, −7.30251873596079134810413331323, −7.17494018187716166597745767980, −6.92527797167722836896630539415, −6.71235103720606587272187125074, −6.49976727006276343069809015171, −6.43932413523730878030676757009, −5.87628588918034514358607270707, −5.86858731207284525319102147790, −5.67051064014435167980319739509, −5.03956062957140599161004051639, −4.51502069935152923555279608600, −4.48208633667020097721719139888, −3.84294579026050579128753428219, −3.70353449986421535336322271474, −3.58290050572778133579737609991, −3.56877360047291224479796960678, −3.04731355933930222633106371832, −2.45944138474087666874750680257, −2.00025551590611070632357829877, −1.77333867030867372193539987598, 0, 0, 0, 0,
1.77333867030867372193539987598, 2.00025551590611070632357829877, 2.45944138474087666874750680257, 3.04731355933930222633106371832, 3.56877360047291224479796960678, 3.58290050572778133579737609991, 3.70353449986421535336322271474, 3.84294579026050579128753428219, 4.48208633667020097721719139888, 4.51502069935152923555279608600, 5.03956062957140599161004051639, 5.67051064014435167980319739509, 5.86858731207284525319102147790, 5.87628588918034514358607270707, 6.43932413523730878030676757009, 6.49976727006276343069809015171, 6.71235103720606587272187125074, 6.92527797167722836896630539415, 7.17494018187716166597745767980, 7.30251873596079134810413331323, 7.76975535155222827325160735093, 7.88174866995798798603224743548, 8.421247875333838184784577305652, 8.561359501563538684735142171081, 8.714251004821748918159059418163
