| L(s) = 1 | − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s + 3·7-s + 8-s + 9-s + 2·10-s − 2·11-s + 2·12-s − 3·14-s + 4·15-s − 16-s + 4·17-s − 18-s + 4·19-s + 2·20-s − 6·21-s + 2·22-s + 10·23-s − 2·24-s − 8·25-s − 3·28-s + 24·29-s − 4·30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s + 0.577·12-s − 0.801·14-s + 1.03·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 1.30·21-s + 0.426·22-s + 2.08·23-s − 0.408·24-s − 8/5·25-s − 0.566·28-s + 4.45·29-s − 0.730·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146371389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146371389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 12 T^{2} + 18 T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 36 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 58 T^{2} - 148 T^{3} + 58 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 70 T^{2} - 324 T^{3} + 70 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 74 T^{2} - 264 T^{3} + 74 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 95 T^{2} + 172 T^{3} + 95 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 62 T^{2} - 208 T^{3} + 62 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 124 T^{2} - 870 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 216 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 868 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 120 T^{2} - 1186 T^{3} + 120 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 172 T^{2} - 66 T^{3} + 172 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 320 T^{2} - 1962 T^{3} + 320 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.847160384926050563372766723910, −8.322969721757107311244260328476, −8.196749684475036403473041425303, −7.72994713740468427557003866366, −7.68340269481159425360959630079, −7.59038657232937249831982601160, −7.00649834005091685907554386497, −6.69755471704903117746792590324, −6.56316894998322927044245583989, −6.11306962711082924718236208244, −5.76073396039833194852667329496, −5.33265368708439148173550385079, −5.32387747564790913349738083003, −4.96319831047848778842086245970, −4.63950150364662894621893567738, −4.34190457560533862622642277021, −4.16416916209665342946348530285, −3.62421914554969948807820759915, −3.15890393527743406937897031922, −2.84268624227887489610977199812, −2.50547037388681242147543823301, −1.93804258008795312668156218110, −1.09467804474640490303257587739, −0.77555690167419617896704934091, −0.67692845902595228387333741118,
0.67692845902595228387333741118, 0.77555690167419617896704934091, 1.09467804474640490303257587739, 1.93804258008795312668156218110, 2.50547037388681242147543823301, 2.84268624227887489610977199812, 3.15890393527743406937897031922, 3.62421914554969948807820759915, 4.16416916209665342946348530285, 4.34190457560533862622642277021, 4.63950150364662894621893567738, 4.96319831047848778842086245970, 5.32387747564790913349738083003, 5.33265368708439148173550385079, 5.76073396039833194852667329496, 6.11306962711082924718236208244, 6.56316894998322927044245583989, 6.69755471704903117746792590324, 7.00649834005091685907554386497, 7.59038657232937249831982601160, 7.68340269481159425360959630079, 7.72994713740468427557003866366, 8.196749684475036403473041425303, 8.322969721757107311244260328476, 8.847160384926050563372766723910
