| L(s) = 1 | + 3·3-s + 2·5-s + 6·9-s + 4·11-s + 3·13-s + 6·15-s + 6·17-s + 25-s + 10·27-s + 6·29-s − 8·31-s + 12·33-s + 2·37-s + 9·39-s + 10·41-s + 4·43-s + 12·45-s − 8·47-s − 5·49-s + 18·51-s + 14·53-s + 8·55-s + 20·59-s + 2·61-s + 6·65-s − 16·67-s − 8·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 2·9-s + 1.20·11-s + 0.832·13-s + 1.54·15-s + 1.45·17-s + 1/5·25-s + 1.92·27-s + 1.11·29-s − 1.43·31-s + 2.08·33-s + 0.328·37-s + 1.44·39-s + 1.56·41-s + 0.609·43-s + 1.78·45-s − 1.16·47-s − 5/7·49-s + 2.52·51-s + 1.92·53-s + 1.07·55-s + 2.60·59-s + 0.256·61-s + 0.744·65-s − 1.95·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.52627183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.52627183\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 41 T^{2} + 16 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 224 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 59 T^{2} - 108 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 143 T^{2} - 812 T^{3} + 143 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 280 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 720 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 171 T^{2} - 1332 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 20 T + 273 T^{2} - 2328 T^{3} + 273 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 137 T^{2} + 1104 T^{3} + 137 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 197 T^{2} + 976 T^{3} + 197 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 327 T^{2} - 3220 T^{3} + 327 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 217 T^{2} - 696 T^{3} + 217 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 159 T^{2} + 852 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 239 T^{2} + 348 T^{3} + 239 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.722042748547116466826453261441, −8.242380823516333586321271767610, −8.234594539876265143727745930925, −8.004754237194569997676362146953, −7.40786604422804926525258339756, −7.30398341204880084912160826061, −7.15848377428148763256013421456, −6.64567124591161293221488038572, −6.37241270030928931515658900528, −6.29353146137862476426063643954, −5.56345335412064371411570317002, −5.51299216643347735925778682239, −5.48836457346978530290082042717, −4.75674760138880651568274604629, −4.26719180503746844960041319043, −4.18275400472922871724009734067, −3.83751670537410676843675557115, −3.53979339416601649404226826960, −3.15597612582686089814662180394, −2.80012352533873832579256761377, −2.50171596612514574494959246919, −2.09798710048113868902423487683, −1.47539273200045269828300646315, −1.36957983056257743767875835517, −0.864284462537430005532481371528,
0.864284462537430005532481371528, 1.36957983056257743767875835517, 1.47539273200045269828300646315, 2.09798710048113868902423487683, 2.50171596612514574494959246919, 2.80012352533873832579256761377, 3.15597612582686089814662180394, 3.53979339416601649404226826960, 3.83751670537410676843675557115, 4.18275400472922871724009734067, 4.26719180503746844960041319043, 4.75674760138880651568274604629, 5.48836457346978530290082042717, 5.51299216643347735925778682239, 5.56345335412064371411570317002, 6.29353146137862476426063643954, 6.37241270030928931515658900528, 6.64567124591161293221488038572, 7.15848377428148763256013421456, 7.30398341204880084912160826061, 7.40786604422804926525258339756, 8.004754237194569997676362146953, 8.234594539876265143727745930925, 8.242380823516333586321271767610, 8.722042748547116466826453261441
