| L(s) = 1 | − 5-s − 4·7-s − 5·11-s + 5·13-s − 2·17-s + 3·19-s − 5·23-s − 4·25-s + 9·29-s + 4·35-s − 6·37-s − 8·41-s − 17·43-s − 47-s − 53-s + 5·55-s − 23·59-s + 3·61-s − 5·65-s − 15·67-s − 12·71-s + 4·73-s + 20·77-s − 26·79-s − 6·83-s + 2·85-s − 18·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.50·11-s + 1.38·13-s − 0.485·17-s + 0.688·19-s − 1.04·23-s − 4/5·25-s + 1.67·29-s + 0.676·35-s − 0.986·37-s − 1.24·41-s − 2.59·43-s − 0.145·47-s − 0.137·53-s + 0.674·55-s − 2.99·59-s + 0.384·61-s − 0.620·65-s − 1.83·67-s − 1.42·71-s + 0.468·73-s + 2.27·77-s − 2.92·79-s − 0.658·83-s + 0.216·85-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $A_4\times C_2$ | \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 4 T + 16 T^{2} + 40 T^{3} + 16 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 5 T + 31 T^{2} + 102 T^{3} + 31 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 5 T + 37 T^{2} - 122 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 42 T^{2} + 66 T^{3} + 42 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 5 T^{2} - 26 T^{3} + 5 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 9 T + 83 T^{2} - 518 T^{3} + 83 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 41 | $A_4\times C_2$ | \( 1 + 8 T + 103 T^{2} + 528 T^{3} + 103 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 17 T + 153 T^{2} + 1094 T^{3} + 153 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + T + 69 T^{2} - 162 T^{3} + 69 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + T + 25 T^{2} - 150 T^{3} + 25 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 23 T + 343 T^{2} + 3090 T^{3} + 343 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 155 T^{2} - 274 T^{3} + 155 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 245 T^{2} + 2042 T^{3} + 245 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 12 T + 137 T^{2} + 776 T^{3} + 137 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 4 T + 152 T^{2} - 258 T^{3} + 152 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 26 T + 421 T^{2} + 4364 T^{3} + 421 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 6 T + 137 T^{2} + 260 T^{3} + 137 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_6$ | \( 1 + 18 T + 251 T^{2} + 2180 T^{3} + 251 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 8 T + 271 T^{2} + 1424 T^{3} + 271 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.369063151249968861609383548655, −7.84496071021368336412911527165, −7.71793018682796446505728006577, −7.64972159253528987301320643714, −7.01917286400862838517889259432, −6.85555895998248098128701314468, −6.79845452247498888318808168777, −6.28347956014771619038934034074, −6.19420992362612097486693823114, −6.07225106409692250627498891888, −5.50575520742664483005463767156, −5.41848660360581299067231283777, −5.16010034541617489477002498200, −4.74946961058206549805289179845, −4.37163683332581426417877666573, −4.25903643338631425718706611270, −3.90779219276774702937557276644, −3.36576331884442475429741780242, −3.29875580263938414283667095713, −3.09097488233587200884320823064, −2.69767407736689287576291242834, −2.56565743892103647066541204691, −1.73745860288119528951036195767, −1.42837358122256925587257562796, −1.38592291591438093795665733606, 0, 0, 0,
1.38592291591438093795665733606, 1.42837358122256925587257562796, 1.73745860288119528951036195767, 2.56565743892103647066541204691, 2.69767407736689287576291242834, 3.09097488233587200884320823064, 3.29875580263938414283667095713, 3.36576331884442475429741780242, 3.90779219276774702937557276644, 4.25903643338631425718706611270, 4.37163683332581426417877666573, 4.74946961058206549805289179845, 5.16010034541617489477002498200, 5.41848660360581299067231283777, 5.50575520742664483005463767156, 6.07225106409692250627498891888, 6.19420992362612097486693823114, 6.28347956014771619038934034074, 6.79845452247498888318808168777, 6.85555895998248098128701314468, 7.01917286400862838517889259432, 7.64972159253528987301320643714, 7.71793018682796446505728006577, 7.84496071021368336412911527165, 8.369063151249968861609383548655
