| L(s) = 1 | + 3·2-s + 6·4-s + 5-s + 7-s + 10·8-s + 3·10-s + 8·11-s − 7·13-s + 3·14-s + 15·16-s + 8·17-s − 9·19-s + 6·20-s + 24·22-s + 8·23-s − 10·25-s − 21·26-s + 6·28-s + 15·29-s + 31-s + 21·32-s + 24·34-s + 35-s + 8·37-s − 27·38-s + 10·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 0.447·5-s + 0.377·7-s + 3.53·8-s + 0.948·10-s + 2.41·11-s − 1.94·13-s + 0.801·14-s + 15/4·16-s + 1.94·17-s − 2.06·19-s + 1.34·20-s + 5.11·22-s + 1.66·23-s − 2·25-s − 4.11·26-s + 1.13·28-s + 2.78·29-s + 0.179·31-s + 3.71·32-s + 4.11·34-s + 0.169·35-s + 1.31·37-s − 4.37·38-s + 1.58·40-s + 0.937·41-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
| \(L(1)\) |
\(\approx\) |
\(36.98304481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(36.98304481\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - T + 11 T^{2} - 7 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 11 T^{2} - 23 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 46 T^{2} - 173 T^{3} + 46 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 7 T + 51 T^{2} + 183 T^{3} + 51 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 4 p T^{2} - 281 T^{3} + 4 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T + 39 T^{2} + 153 T^{3} + 39 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 82 T^{2} - 365 T^{3} + 82 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 15 T + 121 T^{2} - 689 T^{3} + 121 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - T + 23 T^{2} + 179 T^{3} + 23 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 124 T^{2} - 589 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 90 T^{2} - 491 T^{3} + 90 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 835 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 17 p T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 11 T + 131 T^{2} - 1121 T^{3} + 131 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 30 T^{2} - 187 T^{3} + 30 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 93 T^{2} + p T^{3} + 93 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 11 T + 123 T^{2} + 601 T^{3} + 123 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 11 T + 156 T^{2} - 959 T^{3} + 156 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 358 T^{2} - 3687 T^{3} + 358 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 64 T^{2} + 249 T^{3} + 64 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - T + 75 T^{2} - 401 T^{3} + 75 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 201 T^{2} - 1685 T^{3} + 201 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 27 T + 409 T^{2} + 4467 T^{3} + 409 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.52085684260470892820155698236, −6.94271493954001976288838415064, −6.81082526002657178408099756529, −6.79210854074694070035155556312, −6.22157683100491164515028708930, −6.19849815729392799040927836336, −6.16720615912355395508252648343, −5.50774575319953051017029783721, −5.42262476642811331526367429918, −5.38928560999129320156320410696, −4.77233525252531089648499903598, −4.55688410780286419223389743981, −4.45654400327301925554560350208, −4.28212078146229066996560451515, −3.80559035144554312824222778087, −3.78398694233578835851195786748, −3.36668300900673354429355580711, −2.99232050279547211640836497820, −2.71882171667859161812506084598, −2.26917034128382305897727083957, −2.12186680541064327952007146239, −2.08744739881753445409276584616, −1.22126619640319681793905608380, −0.903516770866062483705734914080, −0.864708943727487812510955720255,
0.864708943727487812510955720255, 0.903516770866062483705734914080, 1.22126619640319681793905608380, 2.08744739881753445409276584616, 2.12186680541064327952007146239, 2.26917034128382305897727083957, 2.71882171667859161812506084598, 2.99232050279547211640836497820, 3.36668300900673354429355580711, 3.78398694233578835851195786748, 3.80559035144554312824222778087, 4.28212078146229066996560451515, 4.45654400327301925554560350208, 4.55688410780286419223389743981, 4.77233525252531089648499903598, 5.38928560999129320156320410696, 5.42262476642811331526367429918, 5.50774575319953051017029783721, 6.16720615912355395508252648343, 6.19849815729392799040927836336, 6.22157683100491164515028708930, 6.79210854074694070035155556312, 6.81082526002657178408099756529, 6.94271493954001976288838415064, 7.52085684260470892820155698236
