| L(s) = 1 | − 2-s + 3-s + 2·4-s − 3·5-s − 6-s + 7-s + 2·8-s − 3·9-s + 3·10-s + 2·12-s − 6·13-s − 14-s − 3·15-s − 3·16-s − 4·17-s + 3·18-s − 4·19-s − 6·20-s + 21-s − 6·23-s + 2·24-s + 6·25-s + 6·26-s + 2·28-s − 2·29-s + 3·30-s + 14·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s + 0.707·8-s − 9-s + 0.948·10-s + 0.577·12-s − 1.66·13-s − 0.267·14-s − 0.774·15-s − 3/4·16-s − 0.970·17-s + 0.707·18-s − 0.917·19-s − 1.34·20-s + 0.218·21-s − 1.25·23-s + 0.408·24-s + 6/5·25-s + 1.17·26-s + 0.377·28-s − 0.371·29-s + 0.547·30-s + 2.51·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6602681155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6602681155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T - T^{2} - 5 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - T + 4 T^{2} - 7 T^{3} + 4 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $D_{6}$ | \( 1 - T + 2 T^{2} + 23 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 43 T^{2} + 152 T^{3} + 43 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 88 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 116 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 73 T^{2} + 264 T^{3} + 73 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 59 T^{2} + 44 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 141 T^{2} - 904 T^{3} + 141 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 87 T^{2} - 312 T^{3} + 87 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 78 T^{2} + 441 T^{3} + 78 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 7 T + 126 T^{2} + 539 T^{3} + 126 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 112 T^{2} + 609 T^{3} + 112 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 163 T^{2} + 624 T^{3} + 163 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 832 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 22 T^{2} - 553 T^{3} + 22 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 19 T + 296 T^{2} - 2605 T^{3} + 296 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 217 T^{2} - 840 T^{3} + 217 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 235 T^{2} - 1720 T^{3} + 235 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 161 T^{2} + 20 T^{3} + 161 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 285 T^{2} + 2664 T^{3} + 285 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 11 T + 110 T^{2} - 239 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 63 T^{2} + 1320 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.685688508925631790632117817860, −9.172709997228818673097095227966, −8.630777768741642911386832911604, −8.520359067983594798763783476830, −8.162802840504624379963994081722, −8.107801821397148298845479649277, −7.81719587086280632313461149505, −7.73487137296792916089105770616, −6.97000051294566754405580569451, −6.90075166667405528443125815046, −6.61055564484694900606052599517, −6.46016603035005678001632664064, −5.96050896947728302173885870850, −5.16660466905743938386693024452, −4.97740484161791914639572963685, −4.87777436584057675282077828439, −4.28157916889313020125857102242, −4.14110263773083651934167658110, −3.59261515728408605424602592034, −3.10781789215065694744696590911, −2.63049927980781052070005154559, −2.36553984896459084860831827564, −1.99949519016452146689239223328, −1.35847909486953709471518764152, −0.33924701845223945894802940829,
0.33924701845223945894802940829, 1.35847909486953709471518764152, 1.99949519016452146689239223328, 2.36553984896459084860831827564, 2.63049927980781052070005154559, 3.10781789215065694744696590911, 3.59261515728408605424602592034, 4.14110263773083651934167658110, 4.28157916889313020125857102242, 4.87777436584057675282077828439, 4.97740484161791914639572963685, 5.16660466905743938386693024452, 5.96050896947728302173885870850, 6.46016603035005678001632664064, 6.61055564484694900606052599517, 6.90075166667405528443125815046, 6.97000051294566754405580569451, 7.73487137296792916089105770616, 7.81719587086280632313461149505, 8.107801821397148298845479649277, 8.162802840504624379963994081722, 8.520359067983594798763783476830, 8.630777768741642911386832911604, 9.172709997228818673097095227966, 9.685688508925631790632117817860
