| L(s) = 1 | + 3·2-s + 2·3-s + 4·4-s + 6·6-s + 3·7-s + 4·8-s − 3·9-s − 3·11-s + 8·12-s − 2·13-s + 9·14-s + 3·16-s − 9·18-s + 6·19-s + 6·21-s − 9·22-s + 10·23-s + 8·24-s − 6·26-s − 10·27-s + 12·28-s − 10·29-s − 10·31-s − 32-s − 6·33-s − 12·36-s + 16·37-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.15·3-s + 2·4-s + 2.44·6-s + 1.13·7-s + 1.41·8-s − 9-s − 0.904·11-s + 2.30·12-s − 0.554·13-s + 2.40·14-s + 3/4·16-s − 2.12·18-s + 1.37·19-s + 1.30·21-s − 1.91·22-s + 2.08·23-s + 1.63·24-s − 1.17·26-s − 1.92·27-s + 2.26·28-s − 1.85·29-s − 1.79·31-s − 0.176·32-s − 1.04·33-s − 2·36-s + 2.63·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.15390224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.15390224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 10 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} + 54 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 2 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 220 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 97 T^{2} - 480 T^{3} + 97 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 113 T^{2} + 594 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 16 T + 163 T^{2} - 1084 T^{3} + 163 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 87 T^{2} - 54 T^{3} + 87 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 37 T^{2} - 440 T^{3} + 37 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 20 T + 251 T^{2} - 2038 T^{3} + 251 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 179 T^{2} - 1276 T^{3} + 179 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 1578 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 123 T^{2} - 1282 T^{3} + 123 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 89 T^{2} - 96 T^{3} + 89 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 24 T + 293 T^{2} + 2608 T^{3} + 293 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 394 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 125 T^{2} - 1020 T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 133 T^{2} - 564 T^{3} + 133 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 20 T + 315 T^{2} - 3240 T^{3} + 315 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 99 T^{2} + 160 T^{3} + 99 p T^{4} + p^{3} T^{6} \) |
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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
−8.207814922198991367885121478933, −7.64448083232040295241668296217, −7.49837836788971085122745102977, −7.41354982362207281671582757699, −7.30557719108397494723895141568, −6.98334667970382664708389051177, −6.48834961514351049668363631151, −5.86387710501383565149804319200, −5.73477641198984973467396577916, −5.69859906920156535154015051877, −5.32991593247166286163613919694, −5.23002147620673165970686125272, −4.87906720322584496933308417939, −4.49821885688317793114556483037, −4.25170267823376225133806847421, −4.13128728346611734704593904949, −3.40833533968451430105101986514, −3.37638860017322605653177624788, −3.27871537682490451217339913647, −2.63594584293237975614882500335, −2.41136408610102727067995984034, −2.10880081456513149735554263871, −2.01291905848571577660077671382, −0.851346072241850896268921850035, −0.805156328876437421021000223991,
0.805156328876437421021000223991, 0.851346072241850896268921850035, 2.01291905848571577660077671382, 2.10880081456513149735554263871, 2.41136408610102727067995984034, 2.63594584293237975614882500335, 3.27871537682490451217339913647, 3.37638860017322605653177624788, 3.40833533968451430105101986514, 4.13128728346611734704593904949, 4.25170267823376225133806847421, 4.49821885688317793114556483037, 4.87906720322584496933308417939, 5.23002147620673165970686125272, 5.32991593247166286163613919694, 5.69859906920156535154015051877, 5.73477641198984973467396577916, 5.86387710501383565149804319200, 6.48834961514351049668363631151, 6.98334667970382664708389051177, 7.30557719108397494723895141568, 7.41354982362207281671582757699, 7.49837836788971085122745102977, 7.64448083232040295241668296217, 8.207814922198991367885121478933
