| L(s) = 1 | − 3·5-s + 2·7-s − 7·11-s − 13-s − 10·17-s − 13·19-s − 3·23-s + 6·25-s − 13·29-s − 8·31-s − 6·35-s + 5·37-s − 8·41-s + 24·43-s − 2·47-s − 9·49-s − 53-s + 21·55-s + 17·59-s + 13·61-s + 3·65-s + 5·67-s + 22·71-s + 12·73-s − 14·77-s − 4·79-s + 15·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s − 2.11·11-s − 0.277·13-s − 2.42·17-s − 2.98·19-s − 0.625·23-s + 6/5·25-s − 2.41·29-s − 1.43·31-s − 1.01·35-s + 0.821·37-s − 1.24·41-s + 3.65·43-s − 0.291·47-s − 9/7·49-s − 0.137·53-s + 2.83·55-s + 2.21·59-s + 1.66·61-s + 0.372·65-s + 0.610·67-s + 2.61·71-s + 1.40·73-s − 1.59·77-s − 0.450·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3290769622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3290769622\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 27 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 40 T^{2} + 140 T^{3} + 40 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 16 T^{2} - 24 T^{3} + 16 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 10 T + 61 T^{2} + 257 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 13 T + 90 T^{2} + 432 T^{3} + 90 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 13 T + 113 T^{2} + 678 T^{3} + 113 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 105 T^{2} + 495 T^{3} + 105 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 5 T + 19 T^{2} + 126 T^{3} + 19 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 121 T^{2} + 655 T^{3} + 121 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} - 212 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + T + 59 T^{2} + 406 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 17 T + 243 T^{2} - 2042 T^{3} + 243 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 132 T^{2} - 866 T^{3} + 132 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 109 T^{2} - 174 T^{3} + 109 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 22 T + 309 T^{2} - 2899 T^{3} + 309 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 43 T^{2} + 368 T^{3} + 43 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 120 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 289 T^{2} - 2454 T^{3} + 289 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 139 T^{2} - 1360 T^{3} + 139 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 210 T^{2} + 632 T^{3} + 210 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.05863636150598934177994427581, −6.55913264205417122304155305226, −6.55032331893661723181794685974, −6.41010998119142004645101667310, −5.77672117297107931255997652604, −5.63939587331360595410296630255, −5.63013345753158074281748672730, −5.19332366885526801019576690191, −4.87198410337445255620268498573, −4.83000807713066451680847828344, −4.43618959967067953824648731457, −4.30825237768670758338342987431, −4.04440243182723350194702502142, −3.77338329867495042223101201360, −3.64142121406332749149034107718, −3.43368101085530767956384071677, −2.73133099074177703330270970874, −2.58546679564020879664825248617, −2.23155966342735809752659588312, −2.08182565979903314592751751187, −2.04832530416573198215262039364, −1.71872603270270747772551083490, −0.65559402291503074044996035287, −0.63375782700526148436771818713, −0.14697329336062861772474524143,
0.14697329336062861772474524143, 0.63375782700526148436771818713, 0.65559402291503074044996035287, 1.71872603270270747772551083490, 2.04832530416573198215262039364, 2.08182565979903314592751751187, 2.23155966342735809752659588312, 2.58546679564020879664825248617, 2.73133099074177703330270970874, 3.43368101085530767956384071677, 3.64142121406332749149034107718, 3.77338329867495042223101201360, 4.04440243182723350194702502142, 4.30825237768670758338342987431, 4.43618959967067953824648731457, 4.83000807713066451680847828344, 4.87198410337445255620268498573, 5.19332366885526801019576690191, 5.63013345753158074281748672730, 5.63939587331360595410296630255, 5.77672117297107931255997652604, 6.41010998119142004645101667310, 6.55032331893661723181794685974, 6.55913264205417122304155305226, 7.05863636150598934177994427581
