| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 4·8-s + 8·10-s − 4·11-s + 4·13-s − 12·16-s − 2·17-s + 8·19-s − 8·20-s + 8·22-s + 4·23-s + 10·25-s − 8·26-s − 10·29-s + 4·31-s + 16·32-s + 4·34-s − 16·38-s − 16·40-s − 18·41-s + 8·43-s − 8·44-s − 8·46-s + 20·47-s + 11·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s + 1.41·8-s + 2.52·10-s − 1.20·11-s + 1.10·13-s − 3·16-s − 0.485·17-s + 1.83·19-s − 1.78·20-s + 1.70·22-s + 0.834·23-s + 2·25-s − 1.56·26-s − 1.85·29-s + 0.718·31-s + 2.82·32-s + 0.685·34-s − 2.59·38-s − 2.52·40-s − 2.81·41-s + 1.21·43-s − 1.20·44-s − 1.17·46-s + 2.91·47-s + 11/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9634772773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9634772773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T + p T^{2} )^{2}( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 28 T^{2} - 4 T^{3} + 667 T^{4} - 4 p T^{5} - 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 22 T^{2} - 32 T^{3} + 187 T^{4} - 32 p T^{5} + 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 20 T^{2} + 220 T^{3} + 2659 T^{4} + 220 p T^{5} + 20 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 34 T^{2} - 213 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 4 p T^{2} + 1404 T^{3} + 10635 T^{4} + 1404 p T^{5} + 4 p^{3} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 11 T^{2} + 88 T^{3} + 1024 T^{4} + 88 p T^{5} - 11 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 10 T + 92 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} - 396 T^{3} - 2901 T^{4} - 396 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 59 T^{2} + 88 T^{3} + 6352 T^{4} + 88 p T^{5} - 59 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 36 T^{3} + 14307 T^{4} + 36 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 143 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14 T - 28 T^{2} - 644 T^{3} + 24811 T^{4} - 644 p T^{5} - 28 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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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
−7.65868726144671498418969144983, −7.41112779353487870818943506123, −7.37537593414410839482660703168, −7.12386148751887232090008622154, −7.11739263211705149757404377769, −6.93879215920280144789980605592, −6.40451345474041167073518891434, −5.92117352122135658503789533379, −5.89178104219594047546426532241, −5.48667244980681859187940659312, −5.15227531330021420553006821167, −4.97628378372816097640197134705, −4.94688637499267513545312223204, −4.17086240904881744781820230314, −4.11274320566804161506525350948, −4.02493932765473735818195501659, −3.77627817091040934782069145186, −3.24742990309398207021893285182, −2.92864264914852861520111885156, −2.68091255653233532270575127993, −2.14536226909451793341293445192, −1.89948366494680731156226833154, −1.22104881512354029706243515304, −0.922935228305097979696663115295, −0.53257090231845046999442479275,
0.53257090231845046999442479275, 0.922935228305097979696663115295, 1.22104881512354029706243515304, 1.89948366494680731156226833154, 2.14536226909451793341293445192, 2.68091255653233532270575127993, 2.92864264914852861520111885156, 3.24742990309398207021893285182, 3.77627817091040934782069145186, 4.02493932765473735818195501659, 4.11274320566804161506525350948, 4.17086240904881744781820230314, 4.94688637499267513545312223204, 4.97628378372816097640197134705, 5.15227531330021420553006821167, 5.48667244980681859187940659312, 5.89178104219594047546426532241, 5.92117352122135658503789533379, 6.40451345474041167073518891434, 6.93879215920280144789980605592, 7.11739263211705149757404377769, 7.12386148751887232090008622154, 7.37537593414410839482660703168, 7.41112779353487870818943506123, 7.65868726144671498418969144983
