| L(s) = 1 | − 3·3-s + 3·5-s + 6·9-s − 3·11-s − 2·13-s − 9·15-s − 2·17-s − 8·19-s + 6·25-s − 10·27-s − 10·29-s − 8·31-s + 9·33-s − 6·37-s + 6·39-s − 14·41-s − 4·43-s + 18·45-s + 8·47-s − 5·49-s + 6·51-s − 6·53-s − 9·55-s + 24·57-s − 12·59-s − 6·61-s − 6·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 2·9-s − 0.904·11-s − 0.554·13-s − 2.32·15-s − 0.485·17-s − 1.83·19-s + 6/5·25-s − 1.92·27-s − 1.85·29-s − 1.43·31-s + 1.56·33-s − 0.986·37-s + 0.960·39-s − 2.18·41-s − 0.609·43-s + 2.68·45-s + 1.16·47-s − 5/7·49-s + 0.840·51-s − 0.824·53-s − 1.21·55-s + 3.17·57-s − 1.56·59-s − 0.768·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T - T^{2} - 116 T^{3} - p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 144 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 368 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 14 T + 167 T^{2} + 1156 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 56 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 644 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 161 T^{2} + 1096 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 153 T^{2} - 472 T^{3} + 153 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 181 T^{2} + 1008 T^{3} + 181 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 223 T^{2} + 1700 T^{3} + 223 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 173 T^{2} + 1096 T^{3} + 173 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} - 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 215 T^{2} + 1580 T^{3} + 215 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 399 T^{2} - 4276 T^{3} + 399 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.391663282320161713401167395065, −7.83046392536900667641767918779, −7.70840741781031099767292305126, −7.32247569253705699601316822125, −7.04828671847139615797554412917, −6.89460668168336957054908332059, −6.78014228439698119183105764037, −6.25295378954878238540627991351, −6.07917125953918005816771091669, −5.95722486727299290231293411921, −5.67964388541846028599808024145, −5.34478951113909419184355010781, −5.22847056274548011047854042672, −4.85183502210742771968797599572, −4.58637969334432607644709467657, −4.57138978815205772710479114373, −3.95952266915956602547684779065, −3.56187176924258275913817476086, −3.55370258268752626976725035274, −2.74536603660387801256539091506, −2.65870786203936261835002982436, −2.13306533034983871672222013848, −1.82290899663276006634985141032, −1.46515982737286110635773024591, −1.40744921660663150456257246874, 0, 0, 0,
1.40744921660663150456257246874, 1.46515982737286110635773024591, 1.82290899663276006634985141032, 2.13306533034983871672222013848, 2.65870786203936261835002982436, 2.74536603660387801256539091506, 3.55370258268752626976725035274, 3.56187176924258275913817476086, 3.95952266915956602547684779065, 4.57138978815205772710479114373, 4.58637969334432607644709467657, 4.85183502210742771968797599572, 5.22847056274548011047854042672, 5.34478951113909419184355010781, 5.67964388541846028599808024145, 5.95722486727299290231293411921, 6.07917125953918005816771091669, 6.25295378954878238540627991351, 6.78014228439698119183105764037, 6.89460668168336957054908332059, 7.04828671847139615797554412917, 7.32247569253705699601316822125, 7.70840741781031099767292305126, 7.83046392536900667641767918779, 8.391663282320161713401167395065
