| L(s) = 1 | − 4-s − 3·7-s − 3·8-s − 11-s + 4·13-s − 16-s − 7·17-s + 3·19-s − 23-s + 3·28-s + 29-s − 4·31-s + 6·32-s − 3·37-s − 5·41-s + 22·43-s + 44-s − 10·47-s + 6·49-s − 4·52-s − 19·53-s + 9·56-s + 3·59-s − 8·61-s + 4·64-s − 15·67-s + 7·68-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.13·7-s − 1.06·8-s − 0.301·11-s + 1.10·13-s − 1/4·16-s − 1.69·17-s + 0.688·19-s − 0.208·23-s + 0.566·28-s + 0.185·29-s − 0.718·31-s + 1.06·32-s − 0.493·37-s − 0.780·41-s + 3.35·43-s + 0.150·44-s − 1.45·47-s + 6/7·49-s − 0.554·52-s − 2.60·53-s + 1.20·56-s + 0.390·59-s − 1.02·61-s + 1/2·64-s − 1.83·67-s + 0.848·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} + 3 T^{3} + p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 29 T^{2} + 19 T^{3} + 29 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 34 T^{2} - 105 T^{3} + 34 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 43 T^{2} + 157 T^{3} + 43 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 13 T^{2} - 7 T^{3} + 13 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 61 T^{2} + 43 T^{3} + 61 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T + 79 T^{2} - 55 T^{3} + 79 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 46 T^{2} + 195 T^{3} + 46 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 73 T^{2} + 283 T^{3} + 73 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 13 T^{2} - 271 T^{3} + 13 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 22 T + 286 T^{2} - 2253 T^{3} + 286 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 122 T^{2} + 817 T^{3} + 122 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 19 T + 257 T^{2} + 2161 T^{3} + 257 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 73 T^{2} - 429 T^{3} + 73 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 134 T^{2} + 1025 T^{3} + 134 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 96 T^{2} + 587 T^{3} + 96 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 110 T^{2} - 625 T^{3} + 110 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + T + 138 T^{2} - 135 T^{3} + 138 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - T + 156 T^{2} + 123 T^{3} + 156 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 194 T^{2} + 1369 T^{3} + 194 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 175 T^{2} - 880 T^{3} + 175 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 255 T^{2} + 1644 T^{3} + 255 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.87981219100028227172204537533, −7.38505722648940634223584367728, −7.15221649533612235153521580192, −6.94299011154861351159404329458, −6.71141569573416076029643458610, −6.38131682750270174057663849800, −6.18747723527267889762432671139, −6.13408964004908634789885468252, −5.79931276103549145285478015829, −5.64864079580846044290746332326, −5.11306937717436361672572677257, −4.93557470429255057909127258877, −4.91326204875044506268445858442, −4.20316801294611183965617604339, −4.12352165809774644756744980007, −4.05851549059155064022807055363, −3.50277931449493717046875164951, −3.46462548405289852447864186265, −2.85821174318149857188141486139, −2.78408521181444822526835171788, −2.67033165243441128499796355716, −2.20987536136794925464248301401, −1.66379201495422729046026648013, −1.29408087486877157715752612739, −1.08404170461200630417862878712, 0, 0, 0,
1.08404170461200630417862878712, 1.29408087486877157715752612739, 1.66379201495422729046026648013, 2.20987536136794925464248301401, 2.67033165243441128499796355716, 2.78408521181444822526835171788, 2.85821174318149857188141486139, 3.46462548405289852447864186265, 3.50277931449493717046875164951, 4.05851549059155064022807055363, 4.12352165809774644756744980007, 4.20316801294611183965617604339, 4.91326204875044506268445858442, 4.93557470429255057909127258877, 5.11306937717436361672572677257, 5.64864079580846044290746332326, 5.79931276103549145285478015829, 6.13408964004908634789885468252, 6.18747723527267889762432671139, 6.38131682750270174057663849800, 6.71141569573416076029643458610, 6.94299011154861351159404329458, 7.15221649533612235153521580192, 7.38505722648940634223584367728, 7.87981219100028227172204537533
