| L(s) = 1 | − 3·2-s + 4·4-s + 8·7-s − 4·8-s + 3·11-s + 6·13-s − 24·14-s + 3·16-s − 4·17-s − 2·19-s − 9·22-s − 12·23-s − 18·26-s + 32·28-s + 8·29-s + 8·31-s + 32-s + 12·34-s + 4·37-s + 6·38-s + 8·41-s + 8·43-s + 12·44-s + 36·46-s − 8·47-s + 27·49-s + 24·52-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 3.02·7-s − 1.41·8-s + 0.904·11-s + 1.66·13-s − 6.41·14-s + 3/4·16-s − 0.970·17-s − 0.458·19-s − 1.91·22-s − 2.50·23-s − 3.53·26-s + 6.04·28-s + 1.48·29-s + 1.43·31-s + 0.176·32-s + 2.05·34-s + 0.657·37-s + 0.973·38-s + 1.24·41-s + 1.21·43-s + 1.80·44-s + 5.30·46-s − 1.16·47-s + 27/7·49-s + 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208530079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208530079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 20 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 432 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 624 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 145 T^{2} - 692 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 125 T^{2} + 592 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 127 T^{2} - 576 T^{3} + 127 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 113 T^{2} - 1024 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 185 T^{2} - 1288 T^{3} + 185 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 245 T^{2} - 1688 T^{3} + 245 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 279 T^{2} - 2536 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 245 T^{2} - 328 T^{3} + 245 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 255 T^{2} + 348 T^{3} + 255 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 259 T^{2} + 1424 T^{3} + 259 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.313550274550052274072416963525, −7.71961054666166689607552563894, −7.60933599002272806508545558447, −7.52213522491542998239252820619, −6.96801570104968158819303396632, −6.68238811525108223837190753596, −6.24867010664876310423898326668, −6.23457926262712208796389110711, −6.10320096753296583924939336118, −5.72652633179915019160833579688, −5.11090574914218431648885106409, −4.90640372836009974697442855441, −4.85498820906525652532797636383, −4.22545949719542738448259963100, −4.12566059157906538560690873193, −4.06498697054004920083203930398, −3.52725488187448770204150681972, −3.10559347532886370234082178054, −2.35274810372861076627246468867, −2.18552927336778953146604969676, −2.08670824598092730451087395858, −1.68191823746829392101927104267, −1.08868185482885373761069613152, −0.848118109209223973030578593831, −0.65765036634972510776457339643,
0.65765036634972510776457339643, 0.848118109209223973030578593831, 1.08868185482885373761069613152, 1.68191823746829392101927104267, 2.08670824598092730451087395858, 2.18552927336778953146604969676, 2.35274810372861076627246468867, 3.10559347532886370234082178054, 3.52725488187448770204150681972, 4.06498697054004920083203930398, 4.12566059157906538560690873193, 4.22545949719542738448259963100, 4.85498820906525652532797636383, 4.90640372836009974697442855441, 5.11090574914218431648885106409, 5.72652633179915019160833579688, 6.10320096753296583924939336118, 6.23457926262712208796389110711, 6.24867010664876310423898326668, 6.68238811525108223837190753596, 6.96801570104968158819303396632, 7.52213522491542998239252820619, 7.60933599002272806508545558447, 7.71961054666166689607552563894, 8.313550274550052274072416963525
