| L(s) = 1 | − 6·9-s + 3·11-s + 3·13-s − 6·17-s − 3·19-s − 6·23-s + 27-s − 3·29-s + 15·31-s − 9·37-s − 9·41-s − 3·43-s − 3·47-s − 12·49-s + 3·53-s − 6·59-s − 9·61-s + 3·67-s + 9·71-s + 9·73-s + 12·79-s + 18·81-s − 9·83-s + 6·89-s − 21·97-s − 18·99-s − 12·101-s + ⋯ |
| L(s) = 1 | − 2·9-s + 0.904·11-s + 0.832·13-s − 1.45·17-s − 0.688·19-s − 1.25·23-s + 0.192·27-s − 0.557·29-s + 2.69·31-s − 1.47·37-s − 1.40·41-s − 0.457·43-s − 0.437·47-s − 1.71·49-s + 0.412·53-s − 0.781·59-s − 1.15·61-s + 0.366·67-s + 1.06·71-s + 1.05·73-s + 1.35·79-s + 2·81-s − 0.987·83-s + 0.635·89-s − 2.13·97-s − 1.80·99-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 12 T^{2} - 9 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 27 T^{2} - 49 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 95 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 205 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 187 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 225 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 15 T + 159 T^{2} - 1019 T^{3} + 159 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 9 T + 81 T^{2} + 17 p T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 111 T^{2} + 629 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 120 T^{2} + 239 T^{3} + 120 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 299 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 114 T^{2} - 207 T^{3} + 114 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 412 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 865 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 3 T + 195 T^{2} - 385 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 192 T^{2} - 1097 T^{3} + 192 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 9 T + 237 T^{2} - 1323 T^{3} + 237 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 12 T + 258 T^{2} - 1825 T^{3} + 258 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 9 T + 147 T^{2} + 1205 T^{3} + 147 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 6 T + 258 T^{2} - 1017 T^{3} + 258 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 21 T + 381 T^{2} + 4181 T^{3} + 381 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.18500063598888456280179343810, −6.73721437654338133862118923452, −6.69790770145762566240418162735, −6.67893461360994996013800631602, −6.42905615071169955342807432487, −6.30174949071097421287480648555, −6.00940617421349794564015599381, −5.49493876851529985567375628698, −5.44122795919623231628130919498, −5.42978769200914735716293054032, −4.82693641973352987893287923099, −4.77148524742052573583616099898, −4.48111260346417495366971588174, −3.98181666762047320829862619835, −3.98161085982604781012680334058, −3.82367457930362696189286913767, −3.15736622891091967947916699434, −3.14007727556315150300970911419, −3.06226736499321172121947161278, −2.40098554040902212004215609225, −2.28052406605067301945803821612, −2.13026205379363586518420436118, −1.49693090945598945616003317986, −1.35729470657452941640485665121, −1.01290961884412034345566166753, 0, 0, 0,
1.01290961884412034345566166753, 1.35729470657452941640485665121, 1.49693090945598945616003317986, 2.13026205379363586518420436118, 2.28052406605067301945803821612, 2.40098554040902212004215609225, 3.06226736499321172121947161278, 3.14007727556315150300970911419, 3.15736622891091967947916699434, 3.82367457930362696189286913767, 3.98161085982604781012680334058, 3.98181666762047320829862619835, 4.48111260346417495366971588174, 4.77148524742052573583616099898, 4.82693641973352987893287923099, 5.42978769200914735716293054032, 5.44122795919623231628130919498, 5.49493876851529985567375628698, 6.00940617421349794564015599381, 6.30174949071097421287480648555, 6.42905615071169955342807432487, 6.67893461360994996013800631602, 6.69790770145762566240418162735, 6.73721437654338133862118923452, 7.18500063598888456280179343810
