| L(s) = 1 | + 5-s + 3·7-s + 2·11-s + 5·13-s + 10·17-s + 14·19-s − 5·23-s + 2·25-s + 3·29-s + 7·31-s + 3·35-s − 12·37-s − 12·41-s − 8·43-s − 3·47-s + 8·49-s − 20·53-s + 2·55-s + 14·59-s − 61-s + 5·65-s − 4·67-s − 16·71-s − 14·73-s + 6·77-s − 7·79-s + 25·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.603·11-s + 1.38·13-s + 2.42·17-s + 3.21·19-s − 1.04·23-s + 2/5·25-s + 0.557·29-s + 1.25·31-s + 0.507·35-s − 1.97·37-s − 1.87·41-s − 1.21·43-s − 0.437·47-s + 8/7·49-s − 2.74·53-s + 0.269·55-s + 1.82·59-s − 0.128·61-s + 0.620·65-s − 0.488·67-s − 1.89·71-s − 1.63·73-s + 0.683·77-s − 0.787·79-s + 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.572991797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.572991797\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 18 p T^{5} + p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T + T^{2} + 10 T^{3} + 82 T^{4} + 10 p T^{5} + p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 10 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 28 p T^{5} - 17 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 59 T^{2} + 36 T^{3} - 360 T^{4} + 36 p T^{5} + 59 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 136 p T^{5} - 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 18 p T^{5} - 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 8 p T^{5} - 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 116 p T^{5} - 89 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 28 p T^{5} - 113 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 3700 p T^{5} + 311 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.63977242367379124203311791472, −6.39225880601927591407317536053, −6.33174557438577982036751472002, −5.79957238504614618419366232888, −5.61558287306563807017637004879, −5.54206363425079465280401965759, −5.47914956472277432614705219304, −5.31756264298255985854964911085, −4.83493211487327447488618400362, −4.77454906781457827914668506398, −4.55411492622459458528816860173, −4.12672861421845498903483061222, −4.09711405489823836825393024398, −3.48688238497947763124654640055, −3.40079407867030454196303086427, −3.29083228647024529996166874760, −3.17692316946572100742291090213, −2.73358302946914140227990174579, −2.66148492171291933269431175891, −1.69579933663929288273973558451, −1.59947400193412848163765467526, −1.57649393294833677303553009922, −1.51796376350768886029042874634, −0.76142391979438359876843229339, −0.66204148524340279114253336402,
0.66204148524340279114253336402, 0.76142391979438359876843229339, 1.51796376350768886029042874634, 1.57649393294833677303553009922, 1.59947400193412848163765467526, 1.69579933663929288273973558451, 2.66148492171291933269431175891, 2.73358302946914140227990174579, 3.17692316946572100742291090213, 3.29083228647024529996166874760, 3.40079407867030454196303086427, 3.48688238497947763124654640055, 4.09711405489823836825393024398, 4.12672861421845498903483061222, 4.55411492622459458528816860173, 4.77454906781457827914668506398, 4.83493211487327447488618400362, 5.31756264298255985854964911085, 5.47914956472277432614705219304, 5.54206363425079465280401965759, 5.61558287306563807017637004879, 5.79957238504614618419366232888, 6.33174557438577982036751472002, 6.39225880601927591407317536053, 6.63977242367379124203311791472
