| L(s) = 1 | + 3·5-s − 2·7-s − 4·11-s − 2·13-s − 6·17-s + 2·19-s − 3·23-s + 6·25-s + 2·31-s − 6·35-s + 8·37-s − 2·41-s + 10·43-s − 6·47-s − 6·49-s − 6·53-s − 12·55-s − 8·59-s + 4·61-s − 6·65-s + 8·67-s − 18·71-s + 8·73-s + 8·77-s − 24·79-s − 24·83-s − 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.755·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s − 0.625·23-s + 6/5·25-s + 0.359·31-s − 1.01·35-s + 1.31·37-s − 0.312·41-s + 1.52·43-s − 0.875·47-s − 6/7·49-s − 0.824·53-s − 1.61·55-s − 1.04·59-s + 0.512·61-s − 0.744·65-s + 0.977·67-s − 2.13·71-s + 0.936·73-s + 0.911·77-s − 2.70·79-s − 2.63·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 10 T^{2} + 12 T^{3} + 10 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 52 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 60 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 52 T^{2} + 178 T^{3} + 52 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} - 84 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 76 T^{2} - 12 T^{3} + 76 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 82 T^{2} - 108 T^{3} + 82 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 60 T^{2} - 378 T^{3} + 60 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + 24 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 101 T^{2} - 836 T^{3} + 101 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 372 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 72 T^{2} + 122 T^{3} + 72 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 122 T^{2} + 1016 T^{3} + 122 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 173 T^{2} - 452 T^{3} + 173 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 150 T^{2} - 858 T^{3} + 150 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 310 T^{2} + 2694 T^{3} + 310 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 153 T^{2} - 684 T^{3} + 153 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 24 T + 430 T^{2} + 4420 T^{3} + 430 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 4 T + 147 T^{2} + 136 T^{3} + 147 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 163 T^{2} - 2456 T^{3} + 163 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.31188450261835412725707992558, −6.82181435257892710733971891835, −6.72678815599907300046989428445, −6.64499102970027243840604850828, −6.18474599121711737354252384920, −6.13225864114793256769040514171, −5.97844226930335292282344444067, −5.50053010831972219658789767686, −5.40155735676842053468767319183, −5.34702088235022636661906045224, −4.76706165499783961628323688139, −4.71216434801486076305088088163, −4.58614085990073338678987116275, −4.04840877051860323186689387777, −3.90061333014721204400452020859, −3.82112164509358145366512216445, −2.98504301941135983848302308491, −2.94753007686425542547669341764, −2.92363206255709601629655454951, −2.45105476001722664984402293618, −2.35202821164459295111660446578, −2.10319994057678539398627559737, −1.47123120874092272055658155091, −1.25197726034621787605994447202, −1.18794876188748036836080457074, 0, 0, 0,
1.18794876188748036836080457074, 1.25197726034621787605994447202, 1.47123120874092272055658155091, 2.10319994057678539398627559737, 2.35202821164459295111660446578, 2.45105476001722664984402293618, 2.92363206255709601629655454951, 2.94753007686425542547669341764, 2.98504301941135983848302308491, 3.82112164509358145366512216445, 3.90061333014721204400452020859, 4.04840877051860323186689387777, 4.58614085990073338678987116275, 4.71216434801486076305088088163, 4.76706165499783961628323688139, 5.34702088235022636661906045224, 5.40155735676842053468767319183, 5.50053010831972219658789767686, 5.97844226930335292282344444067, 6.13225864114793256769040514171, 6.18474599121711737354252384920, 6.64499102970027243840604850828, 6.72678815599907300046989428445, 6.82181435257892710733971891835, 7.31188450261835412725707992558
