| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−7.33376090863011107868147408649, −6.91324520176063186936464387081, −6.90169087619487117912239760717, −6.41544173273651734393787299669, −6.36921964877608288052976092361, −6.33302603927934448283351094776, −6.01342761541422454277271179404, −5.65517782601438631137421441792, −5.48023287270713276218560869322, −5.30255266723510980775778256504, −5.15160846394470693014991417812, −4.67647096424725255638775150203, −4.52804143475951001107307018161, −4.19068374196461224466068511198, −3.86022385367553450192566010439, −3.66799842919379848501155809956, −3.30040422528821150970767504078, −3.01194291898326588805269469242, −3.00345646954078469260390831800, −2.52786792820607391260870456151, −2.40903602088920590592348096722, −1.83638364502329388862135036578, −1.60512017687715236486213265871, −1.33945905739843124040361345645, −1.19675062334635903733708696012, 0, 0, 0,
1.19675062334635903733708696012, 1.33945905739843124040361345645, 1.60512017687715236486213265871, 1.83638364502329388862135036578, 2.40903602088920590592348096722, 2.52786792820607391260870456151, 3.00345646954078469260390831800, 3.01194291898326588805269469242, 3.30040422528821150970767504078, 3.66799842919379848501155809956, 3.86022385367553450192566010439, 4.19068374196461224466068511198, 4.52804143475951001107307018161, 4.67647096424725255638775150203, 5.15160846394470693014991417812, 5.30255266723510980775778256504, 5.48023287270713276218560869322, 5.65517782601438631137421441792, 6.01342761541422454277271179404, 6.33302603927934448283351094776, 6.36921964877608288052976092361, 6.41544173273651734393787299669, 6.90169087619487117912239760717, 6.91324520176063186936464387081, 7.33376090863011107868147408649
