| L(s) = 1 | − 3·2-s + 6·4-s − 2·5-s + 3·7-s − 10·8-s + 6·10-s − 3·11-s − 9·14-s + 15·16-s − 2·17-s + 10·19-s − 12·20-s + 9·22-s − 2·23-s − 25-s + 18·28-s + 6·29-s − 21·32-s + 6·34-s − 6·35-s + 10·37-s − 30·38-s + 20·40-s + 2·41-s + 14·43-s − 18·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 0.894·5-s + 1.13·7-s − 3.53·8-s + 1.89·10-s − 0.904·11-s − 2.40·14-s + 15/4·16-s − 0.485·17-s + 2.29·19-s − 2.68·20-s + 1.91·22-s − 0.417·23-s − 1/5·25-s + 3.40·28-s + 1.11·29-s − 3.71·32-s + 1.02·34-s − 1.01·35-s + 1.64·37-s − 4.86·38-s + 3.16·40-s + 0.312·41-s + 2.13·43-s − 2.71·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171575612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171575612\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} + 8 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T^{2} - 16 T^{3} - 5 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 41 T^{2} + 56 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 10 T + 51 T^{2} - 192 T^{3} + 51 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 44 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 55 T^{2} - 252 T^{3} + 55 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 9 T^{2} - 128 T^{3} - 9 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 588 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 113 T^{2} - 152 T^{3} + 113 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 73 T^{2} - 228 T^{3} + 73 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 103 T^{2} - 544 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 179 T^{2} + 1412 T^{3} + 179 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} + 208 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 43 T^{2} + 192 T^{3} + 43 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 149 T^{2} - 472 T^{3} + 149 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2 T + 53 T^{2} + 580 T^{3} + 53 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 273 T^{2} + 2088 T^{3} + 273 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 201 T^{2} - 1120 T^{3} + 201 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 371 T^{2} - 3536 T^{3} + 371 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 295 T^{2} - 2704 T^{3} + 295 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 223 T^{2} - 2524 T^{3} + 223 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.637324549420067304110928291947, −8.043913396954558549893929816633, −7.968853825868422455870902964811, −7.83654600791576800633523905116, −7.45172892337859048678510559791, −7.36429574238662726859879157905, −7.34877622062011195067394977911, −6.73154994663862448635685961520, −6.36516766891233568471330292268, −6.14256330148816180868122592495, −5.70375203652779360508893419870, −5.62023244409812773786000646506, −5.20197464986213116136700492500, −4.65813304757256387640730117019, −4.52504082300354998270121912983, −4.32517181651641376904048294181, −3.48158755369553049833272524310, −3.38761206909651539208914613511, −3.14513020988049816686614276049, −2.36425514081093837518477598391, −2.17156949132925446450463602492, −2.16132774926133614899209265429, −1.11638262699769529792801571224, −0.890362842180685868572167976019, −0.60126124283838711675983718453,
0.60126124283838711675983718453, 0.890362842180685868572167976019, 1.11638262699769529792801571224, 2.16132774926133614899209265429, 2.17156949132925446450463602492, 2.36425514081093837518477598391, 3.14513020988049816686614276049, 3.38761206909651539208914613511, 3.48158755369553049833272524310, 4.32517181651641376904048294181, 4.52504082300354998270121912983, 4.65813304757256387640730117019, 5.20197464986213116136700492500, 5.62023244409812773786000646506, 5.70375203652779360508893419870, 6.14256330148816180868122592495, 6.36516766891233568471330292268, 6.73154994663862448635685961520, 7.34877622062011195067394977911, 7.36429574238662726859879157905, 7.45172892337859048678510559791, 7.83654600791576800633523905116, 7.968853825868422455870902964811, 8.043913396954558549893929816633, 8.637324549420067304110928291947
