| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 3·5-s − 9·6-s − 10·8-s + 6·9-s + 9·10-s + 3·11-s + 18·12-s − 3·13-s − 9·15-s + 15·16-s + 3·17-s − 18·18-s − 6·19-s − 18·20-s − 9·22-s − 6·23-s − 30·24-s − 6·25-s + 9·26-s + 10·27-s + 6·29-s + 27·30-s − 15·31-s − 21·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s − 3.53·8-s + 2·9-s + 2.84·10-s + 0.904·11-s + 5.19·12-s − 0.832·13-s − 2.32·15-s + 15/4·16-s + 0.727·17-s − 4.24·18-s − 1.37·19-s − 4.02·20-s − 1.91·22-s − 1.25·23-s − 6.12·24-s − 6/5·25-s + 1.76·26-s + 1.92·27-s + 1.11·29-s + 4.92·30-s − 2.69·31-s − 3.71·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 11^{3} \cdot 61^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 11^{3} \cdot 61^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 61 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 29 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 18 T^{2} - T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 75 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 99 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 177 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 277 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 297 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 15 T + 129 T^{2} + 771 T^{3} + 129 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 66 T^{2} + 501 T^{3} + 66 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 15 T + 186 T^{2} + 1287 T^{3} + 186 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 407 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 51 T^{2} - 369 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 439 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 9 T + 141 T^{2} + 891 T^{3} + 141 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 27 T + 333 T^{2} + 2915 T^{3} + 333 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 15 T + 225 T^{2} - 2111 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 177 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 473 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 675 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T + 159 T^{2} - 747 T^{3} + 159 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 9 T + 261 T^{2} - 1439 T^{3} + 261 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.026021737622905351765329021845, −7.66263901585514643269519273924, −7.57489780240538482964784239351, −7.38057160405766137241027653003, −6.94463221960050195549748172662, −6.87163496228250944200134569512, −6.72810468015181401284014732446, −6.12388651727663782495362409260, −6.08367649435703144452674966383, −5.97024191619496471751762040766, −5.20919126499081233415548698484, −5.02034904932071385033118301015, −4.89648016076150829914454199365, −4.21626767593644847207337576296, −4.10608850285677161827857621597, −3.83375794872038866067476053526, −3.48947613212434404895955310219, −3.31746901204759266398855489828, −3.28376636387340296217272152613, −2.52881656452819256947787415540, −2.36069895875608784790931203547, −2.07072682259112377398279433552, −1.72285131930852739715669717683, −1.41823600898216194521822303052, −1.26428425200625308885363728382, 0, 0, 0,
1.26428425200625308885363728382, 1.41823600898216194521822303052, 1.72285131930852739715669717683, 2.07072682259112377398279433552, 2.36069895875608784790931203547, 2.52881656452819256947787415540, 3.28376636387340296217272152613, 3.31746901204759266398855489828, 3.48947613212434404895955310219, 3.83375794872038866067476053526, 4.10608850285677161827857621597, 4.21626767593644847207337576296, 4.89648016076150829914454199365, 5.02034904932071385033118301015, 5.20919126499081233415548698484, 5.97024191619496471751762040766, 6.08367649435703144452674966383, 6.12388651727663782495362409260, 6.72810468015181401284014732446, 6.87163496228250944200134569512, 6.94463221960050195549748172662, 7.38057160405766137241027653003, 7.57489780240538482964784239351, 7.66263901585514643269519273924, 8.026021737622905351765329021845
