| L(s) = 1 | − 3·5-s + 3·7-s − 3·9-s + 3·11-s − 3·17-s + 3·19-s + 3·23-s + 6·25-s + 3·27-s + 3·29-s + 6·31-s − 9·35-s + 12·37-s − 6·41-s + 18·43-s + 9·45-s + 15·47-s − 9·49-s + 6·53-s − 9·55-s + 6·59-s + 27·61-s − 9·63-s + 12·67-s + 6·71-s − 15·73-s + 9·77-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s − 9-s + 0.904·11-s − 0.727·17-s + 0.688·19-s + 0.625·23-s + 6/5·25-s + 0.577·27-s + 0.557·29-s + 1.07·31-s − 1.52·35-s + 1.97·37-s − 0.937·41-s + 2.74·43-s + 1.34·45-s + 2.18·47-s − 9/7·49-s + 0.824·53-s − 1.21·55-s + 0.781·59-s + 3.45·61-s − 1.13·63-s + 1.46·67-s + 0.712·71-s − 1.75·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.651862107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.651862107\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} - p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} - 40 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} - 78 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 36 T^{2} + 114 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 24 T^{2} - 162 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 81 T^{2} - 162 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 123 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 423 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 18 T + 201 T^{2} - 1516 T^{3} + 201 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 15 T + 207 T^{2} - 1486 T^{3} + 207 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 27 T^{2} - 100 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 141 T^{2} - 676 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 27 T + 408 T^{2} - 3890 T^{3} + 408 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T - 39 T^{2} + 1336 T^{3} - 39 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 153 T^{2} - 649 T^{3} + 153 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1602 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 189 T^{2} + 1928 T^{3} + 189 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 531 T^{2} + 5948 T^{3} + 531 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 222 T^{2} - 1608 T^{3} + 222 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.845705517746448356968639695938, −8.549828905158112225631934999399, −8.476557987426808450293619736966, −8.227255071204510559461661522701, −7.78037878588799609529744487742, −7.74931592723668786958483191954, −7.14021441660917563835613240358, −7.10423350666198208589093060925, −6.65038450316429012628623732678, −6.60294314336283483139283473760, −5.90017753193875975243790397177, −5.64340705669167632379914834182, −5.60191294919404810413700891064, −4.95667437990707476866512516339, −4.66439229664051680915013797051, −4.48558544087594275423064035637, −4.11741112209726206191038807140, −3.69275157339682111977888467977, −3.64365954942181291161219536478, −2.73648436186003824146470648247, −2.58051813204220411644192487717, −2.52431916908950060007216188222, −1.52725803079340913300722093006, −0.925421902208183046618648672404, −0.71616426428189796732043876617,
0.71616426428189796732043876617, 0.925421902208183046618648672404, 1.52725803079340913300722093006, 2.52431916908950060007216188222, 2.58051813204220411644192487717, 2.73648436186003824146470648247, 3.64365954942181291161219536478, 3.69275157339682111977888467977, 4.11741112209726206191038807140, 4.48558544087594275423064035637, 4.66439229664051680915013797051, 4.95667437990707476866512516339, 5.60191294919404810413700891064, 5.64340705669167632379914834182, 5.90017753193875975243790397177, 6.60294314336283483139283473760, 6.65038450316429012628623732678, 7.10423350666198208589093060925, 7.14021441660917563835613240358, 7.74931592723668786958483191954, 7.78037878588799609529744487742, 8.227255071204510559461661522701, 8.476557987426808450293619736966, 8.549828905158112225631934999399, 8.845705517746448356968639695938
