| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 5-s + 2·6-s − 3·7-s − 4·8-s − 2·9-s − 2·10-s − 3·12-s − 8·13-s + 6·14-s − 15-s + 3·16-s − 8·17-s + 4·18-s + 3·20-s + 3·21-s + 7·23-s + 4·24-s − 6·25-s + 16·26-s − 3·27-s − 9·28-s + 2·30-s − 13·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s − 1.13·7-s − 1.41·8-s − 2/3·9-s − 0.632·10-s − 0.866·12-s − 2.21·13-s + 1.60·14-s − 0.258·15-s + 3/4·16-s − 1.94·17-s + 0.942·18-s + 0.670·20-s + 0.654·21-s + 1.45·23-s + 0.816·24-s − 6/5·25-s + 3.13·26-s − 0.577·27-s − 1.70·28-s + 0.365·30-s − 2.33·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 7 T^{2} - 6 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 144 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 208 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 64 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 77 T^{2} - 314 T^{3} + 77 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 47 T^{2} - 64 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 13 T + 143 T^{2} + 864 T^{3} + 143 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 17 T + 183 T^{2} + 1274 T^{3} + 183 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 189 T^{2} + 1392 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 101 T^{2} + 312 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 127 T^{2} - 384 T^{3} + 127 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 3 p T^{2} + 996 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - T + 171 T^{2} - 120 T^{3} + 171 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 16 T + 249 T^{2} - 2032 T^{3} + 249 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 113 T^{2} - 22 T^{3} + 113 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 5 T + 5 T^{2} + 770 T^{3} + 5 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 16 T + 285 T^{2} + 2416 T^{3} + 285 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 28 T + 465 T^{2} + 4936 T^{3} + 465 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 193 T^{2} + 1392 T^{3} + 193 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 21 T + 371 T^{2} + 3838 T^{3} + 371 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 11 T + 259 T^{2} + 1682 T^{3} + 259 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.614463791621845598751894058632, −9.323969163944463819598812375727, −8.829655988538240971022437382686, −8.626730733833483058977127668330, −8.592182756996856171849017779293, −8.442927201611685030658396091008, −7.53741827075668979669493183470, −7.32708875773608493986839165494, −7.21368767281873072891819877104, −6.97234293506777107113999358625, −6.84785483325004550285318440340, −6.47134988274745794866535561096, −5.92419631713016693379617853779, −5.75798929761205348688624068085, −5.38532332212894613047907448626, −5.26173842224527021115874685524, −4.72407829068804524491932456627, −4.49830365428864379742761539316, −3.66764501922848674944150779463, −3.61730994243451896278724207935, −3.03696002478874658447951605703, −2.74035320139465276450702291961, −2.14632884657505271210767399285, −1.90596557038024625654926636672, −1.64693816130346012129633270933, 0, 0, 0,
1.64693816130346012129633270933, 1.90596557038024625654926636672, 2.14632884657505271210767399285, 2.74035320139465276450702291961, 3.03696002478874658447951605703, 3.61730994243451896278724207935, 3.66764501922848674944150779463, 4.49830365428864379742761539316, 4.72407829068804524491932456627, 5.26173842224527021115874685524, 5.38532332212894613047907448626, 5.75798929761205348688624068085, 5.92419631713016693379617853779, 6.47134988274745794866535561096, 6.84785483325004550285318440340, 6.97234293506777107113999358625, 7.21368767281873072891819877104, 7.32708875773608493986839165494, 7.53741827075668979669493183470, 8.442927201611685030658396091008, 8.592182756996856171849017779293, 8.626730733833483058977127668330, 8.829655988538240971022437382686, 9.323969163944463819598812375727, 9.614463791621845598751894058632
