| L(s) = 1 | + 2·2-s + 4-s − 2·5-s + 4·7-s − 2·8-s − 4·10-s − 4·11-s + 2·13-s + 8·14-s − 4·16-s + 12·19-s − 2·20-s − 8·22-s − 2·23-s + 8·25-s + 4·26-s + 4·28-s − 2·29-s + 4·31-s − 2·32-s − 8·35-s − 24·37-s + 24·38-s + 4·40-s − 18·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 0.707·8-s − 1.26·10-s − 1.20·11-s + 0.554·13-s + 2.13·14-s − 16-s + 2.75·19-s − 0.447·20-s − 1.70·22-s − 0.417·23-s + 8/5·25-s + 0.784·26-s + 0.755·28-s − 0.371·29-s + 0.718·31-s − 0.353·32-s − 1.35·35-s − 3.94·37-s + 3.89·38-s + 0.632·40-s − 2.81·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5260259247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5260259247\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T - 4 T^{2} - 4 T^{3} + 19 T^{4} - 4 p T^{5} - 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + T^{2} - 4 T^{3} + 64 T^{4} - 4 p T^{5} + p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 7 T^{2} + 4 T^{3} + 232 T^{4} + 4 p T^{5} - 7 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 52 T^{3} - 221 T^{4} - 52 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 22 T^{3} + 1252 T^{4} - 22 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 38 T^{2} + 32 T^{3} + 1459 T^{4} + 32 p T^{5} - 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 4 T - 26 T^{2} + 176 T^{3} - 773 T^{4} + 176 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 110 T^{2} + 832 T^{3} + 7075 T^{4} + 832 p T^{5} + 110 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 67 T^{2} + 104 T^{3} + 9904 T^{4} + 104 p T^{5} - 67 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T - 47 T^{2} - 88 T^{3} + 4696 T^{4} - 88 p T^{5} - 47 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 22 T^{2} - 736 T^{3} - 7013 T^{4} - 736 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 155 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T - 128 T^{2} + 52 T^{3} + 10867 T^{4} + 52 p T^{5} - 128 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 - 163 T^{2} + 19680 T^{4} - 163 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 160 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 10 T - 116 T^{2} + 220 T^{3} + 27547 T^{4} + 220 p T^{5} - 116 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.65495605911808078002642626910, −6.05505466283440958196458853658, −5.78352058987840913788428577415, −5.70287610151473388827034524293, −5.49504355795238319256167123954, −5.39148317235382564909973808226, −5.04566340429515439781577502537, −4.94221495514448330439039268979, −4.89717722135159697096672964178, −4.75777179193384338369110591872, −4.42104189111532090502491692338, −3.94586966352482256103471520908, −3.81579806839292712127108358467, −3.76438198009230223649727214923, −3.36680999624882043998214294260, −3.27389088621332962054789077218, −3.15778854311640775513978274279, −2.67607569047536963884230594186, −2.50950836345299019447686592868, −2.06104523704225003684701285957, −1.94309141310677753050423118824, −1.39156657826037108155557357547, −1.14509969527452040214592027751, −0.948906439198902314950339057350, −0.093154557555477989788543417803,
0.093154557555477989788543417803, 0.948906439198902314950339057350, 1.14509969527452040214592027751, 1.39156657826037108155557357547, 1.94309141310677753050423118824, 2.06104523704225003684701285957, 2.50950836345299019447686592868, 2.67607569047536963884230594186, 3.15778854311640775513978274279, 3.27389088621332962054789077218, 3.36680999624882043998214294260, 3.76438198009230223649727214923, 3.81579806839292712127108358467, 3.94586966352482256103471520908, 4.42104189111532090502491692338, 4.75777179193384338369110591872, 4.89717722135159697096672964178, 4.94221495514448330439039268979, 5.04566340429515439781577502537, 5.39148317235382564909973808226, 5.49504355795238319256167123954, 5.70287610151473388827034524293, 5.78352058987840913788428577415, 6.05505466283440958196458853658, 6.65495605911808078002642626910
