| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·5-s + 4·6-s − 3·7-s + 2·8-s + 9-s − 8·10-s − 2·12-s − 13-s + 6·14-s − 8·15-s − 4·16-s − 2·17-s − 2·18-s + 3·19-s + 4·20-s + 6·21-s + 3·23-s − 4·24-s + 10·25-s + 2·26-s + 2·27-s − 3·28-s − 9·29-s + 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.78·5-s + 1.63·6-s − 1.13·7-s + 0.707·8-s + 1/3·9-s − 2.52·10-s − 0.577·12-s − 0.277·13-s + 1.60·14-s − 2.06·15-s − 16-s − 0.485·17-s − 0.471·18-s + 0.688·19-s + 0.894·20-s + 1.30·21-s + 0.625·23-s − 0.816·24-s + 2·25-s + 0.392·26-s + 0.384·27-s − 0.566·28-s − 1.67·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6617512566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6617512566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 3 T - 3 T^{2} - 6 T^{3} + 32 T^{4} - 6 p T^{5} - 3 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 14 T^{2} - 32 T^{3} - 33 T^{4} - 32 p T^{5} - 14 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 27 T^{2} + 6 T^{3} + 764 T^{4} + 6 p T^{5} - 27 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - T^{2} + 108 T^{3} - 636 T^{4} + 108 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 9 T + 7 T^{2} + 144 T^{3} + 2286 T^{4} + 144 p T^{5} + 7 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 57 T^{2} + 1880 T^{4} - 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 8 T + 34 T^{2} - 416 T^{3} - 3405 T^{4} - 416 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 63 T^{2} + 10 T^{3} + 4820 T^{4} + 10 p T^{5} - 63 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 17 T + 103 T^{2} + 1156 T^{3} + 14064 T^{4} + 1156 p T^{5} + 103 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 42 T^{2} - 384 T^{3} - 3733 T^{4} - 384 p T^{5} + 42 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 118 T^{2} - 1152 T^{3} + 14391 T^{4} - 1152 p T^{5} + 118 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 19 T + 244 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 17 T + 145 T^{2} + 578 T^{3} - 12906 T^{4} + 578 p T^{5} + 145 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 6 T - 150 T^{2} + 48 T^{3} + 21695 T^{4} + 48 p T^{5} - 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.299299386119280621650314120078, −7.936950627451422516493769393711, −7.66115662753250876686988131040, −7.42285120217096377999195807512, −7.22766625826689786359455255311, −6.89118418684099624412996700709, −6.71074590948572232831897638643, −6.60617696497694105883191654303, −6.11221232787929381586546860074, −5.85779682163974014792744334163, −5.71596949922371053169128149454, −5.50569328611248731942388239493, −5.27542527390200184001034118722, −5.04728888401419224601820863295, −4.46075643872445643519425381494, −4.43809435009555833113135851153, −3.81911376440310712694617052163, −3.64285917120525226684542045420, −3.11629016998815189811666501121, −2.76576603052525779819467519888, −2.39061168634491868057978214172, −1.85435257115306687980002757043, −1.78480068823824731481418273688, −0.77717718719313302481954487118, −0.65060165622980218014313064976,
0.65060165622980218014313064976, 0.77717718719313302481954487118, 1.78480068823824731481418273688, 1.85435257115306687980002757043, 2.39061168634491868057978214172, 2.76576603052525779819467519888, 3.11629016998815189811666501121, 3.64285917120525226684542045420, 3.81911376440310712694617052163, 4.43809435009555833113135851153, 4.46075643872445643519425381494, 5.04728888401419224601820863295, 5.27542527390200184001034118722, 5.50569328611248731942388239493, 5.71596949922371053169128149454, 5.85779682163974014792744334163, 6.11221232787929381586546860074, 6.60617696497694105883191654303, 6.71074590948572232831897638643, 6.89118418684099624412996700709, 7.22766625826689786359455255311, 7.42285120217096377999195807512, 7.66115662753250876686988131040, 7.936950627451422516493769393711, 8.299299386119280621650314120078
