| L(s) = 1 | + 3·2-s + 3-s + 6·4-s + 5-s + 3·6-s − 3·7-s + 10·8-s + 3·10-s + 6·11-s + 6·12-s − 9·14-s + 15-s + 15·16-s − 7·17-s + 4·19-s + 6·20-s − 3·21-s + 18·22-s + 4·23-s + 10·24-s − 8·25-s + 27-s − 18·28-s + 5·29-s + 3·30-s + 7·31-s + 21·32-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 3·4-s + 0.447·5-s + 1.22·6-s − 1.13·7-s + 3.53·8-s + 0.948·10-s + 1.80·11-s + 1.73·12-s − 2.40·14-s + 0.258·15-s + 15/4·16-s − 1.69·17-s + 0.917·19-s + 1.34·20-s − 0.654·21-s + 3.83·22-s + 0.834·23-s + 2.04·24-s − 8/5·25-s + 0.192·27-s − 3.40·28-s + 0.928·29-s + 0.547·30-s + 1.25·31-s + 3.71·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.25917625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.25917625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + T^{2} - 2 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 9 T^{2} - 12 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 112 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 59 T^{2} + 230 T^{3} + 59 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 29 T^{2} - 120 T^{3} + 29 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 89 T^{2} - 280 T^{3} + 89 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 7 T + 101 T^{2} - 426 T^{3} + 101 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 81 T^{2} + 542 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 + 7 T + 113 T^{2} + 632 T^{3} + 113 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 127 T^{2} + 336 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 13 T + 209 T^{2} + 1596 T^{3} + 209 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 22 T + 267 T^{2} - 2368 T^{3} + 267 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 7 T + 109 T^{2} - 666 T^{3} + 109 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 28 T + 447 T^{2} + 4600 T^{3} + 447 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 19 T + 333 T^{2} - 3130 T^{3} + 333 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 295 T^{2} + 2304 T^{3} + 295 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 23 T + 419 T^{2} - 4330 T^{3} + 419 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 25 T + 475 T^{2} + 5254 T^{3} + 475 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.671194476078286043008974889804, −9.229426680917085371189358931732, −9.128817925669330731053712170305, −8.780395448069621168105909460393, −8.182591854351282564389236979900, −8.129630328776485933218086692755, −7.63001797093701967215520971330, −7.19779991738205797877288802825, −6.89772780772444763321622178266, −6.64596758441776910553999610122, −6.36778726797042109542187846531, −6.19769143318751476306815788592, −6.04217340344211938162733859461, −5.30799897326736813434100659852, −5.19702182612899164574723190474, −4.73215520389938805255920608766, −4.27948145793271618732746558621, −4.00608475745588460458165999242, −3.90409143678390936851522689720, −3.21102121982046635593389337956, −2.90011562338123805775820843040, −2.84344174822318155315235928310, −2.05396122224742170942447373918, −1.75206455187883530280281519592, −0.987019722953690538217866446099,
0.987019722953690538217866446099, 1.75206455187883530280281519592, 2.05396122224742170942447373918, 2.84344174822318155315235928310, 2.90011562338123805775820843040, 3.21102121982046635593389337956, 3.90409143678390936851522689720, 4.00608475745588460458165999242, 4.27948145793271618732746558621, 4.73215520389938805255920608766, 5.19702182612899164574723190474, 5.30799897326736813434100659852, 6.04217340344211938162733859461, 6.19769143318751476306815788592, 6.36778726797042109542187846531, 6.64596758441776910553999610122, 6.89772780772444763321622178266, 7.19779991738205797877288802825, 7.63001797093701967215520971330, 8.129630328776485933218086692755, 8.182591854351282564389236979900, 8.780395448069621168105909460393, 9.128817925669330731053712170305, 9.229426680917085371189358931732, 9.671194476078286043008974889804
