| L(s) = 1 | − 3·2-s + 7·4-s − 6·5-s + 2·7-s − 14·8-s + 18·10-s − 5·11-s − 6·14-s + 21·16-s + 17-s − 7·19-s − 42·20-s + 15·22-s + 16·25-s + 14·28-s + 2·29-s − 16·31-s − 28·32-s − 3·34-s − 12·35-s + 22·37-s + 21·38-s + 84·40-s − 11·41-s − 15·43-s − 35·44-s − 7·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 7/2·4-s − 2.68·5-s + 0.755·7-s − 4.94·8-s + 5.69·10-s − 1.50·11-s − 1.60·14-s + 21/4·16-s + 0.242·17-s − 1.60·19-s − 9.39·20-s + 3.19·22-s + 16/5·25-s + 2.64·28-s + 0.371·29-s − 2.87·31-s − 4.94·32-s − 0.514·34-s − 2.02·35-s + 3.61·37-s + 3.40·38-s + 13.2·40-s − 1.71·41-s − 2.28·43-s − 5.27·44-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_6$ | \( 1 + 3 T + p T^{2} - T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 47 T^{3} + 4 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 2 T + 20 T^{2} - 27 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 5 T + 25 T^{2} + 69 T^{3} + 25 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - T + 35 T^{2} - 47 T^{3} + 35 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 7 T + 71 T^{2} + 273 T^{3} + 71 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 20 T^{2} - 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 2 T + 72 T^{2} - 3 p T^{3} + 72 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 16 T + 134 T^{2} + 795 T^{3} + 134 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 22 T + 270 T^{2} - 2005 T^{3} + 270 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 11 T + 147 T^{2} + 873 T^{3} + 147 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 15 T + 176 T^{2} + 1331 T^{3} + 176 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 7 T + 155 T^{2} + 665 T^{3} + 155 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 17 T + 225 T^{2} - 1761 T^{3} + 225 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 161 T^{2} - 604 T^{3} + 161 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 11 T + 155 T^{2} - 1515 T^{3} + 155 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 122 T^{2} - 203 T^{3} + 122 p T^{4} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 84 T^{2} + 47 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 12 T + 290 T^{2} + 2035 T^{3} + 290 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + T + 167 T^{2} + 65 T^{3} + 167 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 10 T^{2} - 667 T^{3} + 10 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.729046502714617961673117664127, −8.454102009110094847811934302323, −8.072609369527105925927436666143, −8.048507616854019405875601541002, −7.974127885516665872008237827783, −7.59135585279195456593918061371, −7.46538941602813765018438998805, −7.02428258241978688927199821919, −6.94357930852440145570557060532, −6.49573639144124491397065658889, −6.34468635607158477440391547400, −6.04570218867001902799632484874, −5.39666728048454627452482890341, −5.29259027200472378905888742564, −4.75887433727182321101608125283, −4.68521947225991398990664191448, −4.11068676397083341926983372422, −3.81091716886499003800861333478, −3.50442496680246461454648537459, −3.24565746953868449761583033097, −2.68803514739655479749256292802, −2.34558679142852012484468111324, −2.20967370075094653944469815960, −1.52304287975092734240630776917, −1.16219322496525216282147802945, 0, 0, 0,
1.16219322496525216282147802945, 1.52304287975092734240630776917, 2.20967370075094653944469815960, 2.34558679142852012484468111324, 2.68803514739655479749256292802, 3.24565746953868449761583033097, 3.50442496680246461454648537459, 3.81091716886499003800861333478, 4.11068676397083341926983372422, 4.68521947225991398990664191448, 4.75887433727182321101608125283, 5.29259027200472378905888742564, 5.39666728048454627452482890341, 6.04570218867001902799632484874, 6.34468635607158477440391547400, 6.49573639144124491397065658889, 6.94357930852440145570557060532, 7.02428258241978688927199821919, 7.46538941602813765018438998805, 7.59135585279195456593918061371, 7.974127885516665872008237827783, 8.048507616854019405875601541002, 8.072609369527105925927436666143, 8.454102009110094847811934302323, 8.729046502714617961673117664127
