| L(s) = 1 | + 3·2-s + 6·4-s + 5-s + 9·7-s + 10·8-s + 3·10-s − 6·11-s + 11·13-s + 27·14-s + 15·16-s + 2·17-s + 5·19-s + 6·20-s − 18·22-s + 2·23-s − 6·25-s + 33·26-s + 54·28-s + 9·29-s − 5·31-s + 21·32-s + 6·34-s + 9·35-s + 4·37-s + 15·38-s + 10·40-s − 8·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 0.447·5-s + 3.40·7-s + 3.53·8-s + 0.948·10-s − 1.80·11-s + 3.05·13-s + 7.21·14-s + 15/4·16-s + 0.485·17-s + 1.14·19-s + 1.34·20-s − 3.83·22-s + 0.417·23-s − 6/5·25-s + 6.47·26-s + 10.2·28-s + 1.67·29-s − 0.898·31-s + 3.71·32-s + 1.02·34-s + 1.52·35-s + 0.657·37-s + 2.43·38-s + 1.58·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 223^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 223^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(55.50573898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(55.50573898\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 223 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - T + 7 T^{2} + T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 9 T + 43 T^{2} - 137 T^{3} + 43 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 40 T^{2} + 129 T^{3} + 40 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 11 T + 71 T^{2} - 313 T^{3} + 71 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 30 T^{2} - 13 T^{3} + 30 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 5 T + 55 T^{2} - 175 T^{3} + 55 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 38 T^{2} - 17 T^{3} + 38 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 9 T + 109 T^{2} - 535 T^{3} + 109 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 57 T^{2} + 141 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T - 10 T^{2} + 413 T^{3} - 10 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 26 T^{2} + 101 T^{3} + 26 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - p T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 11 T + 147 T^{2} - 881 T^{3} + 147 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 91 T^{2} + 29 T^{3} + 91 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 114 T^{2} - 789 T^{3} + 114 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + T + 55 T^{2} - 463 T^{3} + 55 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 15 T + 3 p T^{2} - 1537 T^{3} + 3 p^{2} T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 15 T + 268 T^{2} - 2163 T^{3} + 268 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 54 T^{2} + 705 T^{3} + 54 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 22 T + 270 T^{2} - 2387 T^{3} + 270 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1491 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T - 17 T^{2} - 1617 T^{3} - 17 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + T + 173 T^{2} + 413 T^{3} + 173 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.64712102595978213019073337404, −7.21241186624343474639211307303, −6.92429505841446988530751247903, −6.67850694435423762159399420546, −6.21305099496710156157566539514, −6.00339568473177596723570548557, −5.98685420476351020406840193754, −5.40382253388563416305565167426, −5.38911737832299879927137913686, −5.34510008028523932964568574840, −4.93727995293465074691236853050, −4.66912478080721850522319444198, −4.64542513836203150636448811299, −4.20257617562639338707166730792, −3.78599994495972300135339412166, −3.64616715727597076811049171137, −3.49162162842925893987519108614, −3.02732564308508361547906385761, −2.71672351326373620610274619553, −2.16747218917727764414918443977, −2.02939162692101070573620563833, −1.99276696267867831427129386799, −1.17828950136303854496352532166, −1.17033166431240047115197318848, −0.941916445829507606700782102901,
0.941916445829507606700782102901, 1.17033166431240047115197318848, 1.17828950136303854496352532166, 1.99276696267867831427129386799, 2.02939162692101070573620563833, 2.16747218917727764414918443977, 2.71672351326373620610274619553, 3.02732564308508361547906385761, 3.49162162842925893987519108614, 3.64616715727597076811049171137, 3.78599994495972300135339412166, 4.20257617562639338707166730792, 4.64542513836203150636448811299, 4.66912478080721850522319444198, 4.93727995293465074691236853050, 5.34510008028523932964568574840, 5.38911737832299879927137913686, 5.40382253388563416305565167426, 5.98685420476351020406840193754, 6.00339568473177596723570548557, 6.21305099496710156157566539514, 6.67850694435423762159399420546, 6.92429505841446988530751247903, 7.21241186624343474639211307303, 7.64712102595978213019073337404
