| L(s) = 1 | − 3·2-s + 6·4-s − 6·5-s + 3·7-s − 10·8-s − 6·9-s + 18·10-s + 9·11-s − 9·13-s − 9·14-s + 15·16-s − 9·17-s + 18·18-s − 3·19-s − 36·20-s − 27·22-s + 15·23-s + 12·25-s + 27·26-s − 27-s + 18·28-s − 9·31-s − 21·32-s + 27·34-s − 18·35-s − 36·36-s + 9·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 2.68·5-s + 1.13·7-s − 3.53·8-s − 2·9-s + 5.69·10-s + 2.71·11-s − 2.49·13-s − 2.40·14-s + 15/4·16-s − 2.18·17-s + 4.24·18-s − 0.688·19-s − 8.04·20-s − 5.75·22-s + 3.12·23-s + 12/5·25-s + 5.29·26-s − 0.192·27-s + 3.40·28-s − 1.61·31-s − 3.71·32-s + 4.63·34-s − 3.04·35-s − 6·36-s + 1.45·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 37 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 61 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 39 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 9 T + 57 T^{2} - 215 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 9 T + 57 T^{2} + 243 T^{3} + 57 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 61 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 15 T + 123 T^{2} - 693 T^{3} + 123 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 24 T^{2} - 171 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 575 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 36 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 6 T + 48 T^{2} - 49 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 96 T^{2} - 299 T^{3} + 96 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 123 T^{2} + 963 T^{3} + 123 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} + 475 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1400 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 21 T + 345 T^{2} - 3135 T^{3} + 345 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 24 T + 321 T^{2} - 2952 T^{3} + 321 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 18 T + 306 T^{2} + 2681 T^{3} + 306 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 156 T^{2} + 687 T^{3} + 156 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 9 T + 237 T^{2} + 1315 T^{3} + 237 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} + 295 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 42 T + 867 T^{2} + 10716 T^{3} + 867 p T^{4} + 42 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.382235023745211916485948384476, −8.071886781872829451147353063555, −7.84174519173191008412418554691, −7.43872689938246279510272419550, −7.14739672886643637708238059648, −7.11175493971575409100580667568, −7.05796877467521089360941920293, −6.57362204860692790572615119054, −6.39520192937732391147745376162, −6.22990687195589009198959786963, −5.54729117874674327735416141522, −5.29186936717760212854733173144, −5.13824647734493018860056387016, −4.60072818648484320759530004622, −4.41829537392196869356612662965, −4.35478744899301695462912153340, −3.62622847306803606074184222661, −3.55815702541249852611913928600, −3.42137522488838847047268610632, −2.62321141435390868111492291601, −2.57301212437560185704986470157, −2.38197258005475929023625933147, −1.56095544951989236992314174643, −1.54466705362606722833802152926, −0.909364769270692723370769713133, 0, 0, 0,
0.909364769270692723370769713133, 1.54466705362606722833802152926, 1.56095544951989236992314174643, 2.38197258005475929023625933147, 2.57301212437560185704986470157, 2.62321141435390868111492291601, 3.42137522488838847047268610632, 3.55815702541249852611913928600, 3.62622847306803606074184222661, 4.35478744899301695462912153340, 4.41829537392196869356612662965, 4.60072818648484320759530004622, 5.13824647734493018860056387016, 5.29186936717760212854733173144, 5.54729117874674327735416141522, 6.22990687195589009198959786963, 6.39520192937732391147745376162, 6.57362204860692790572615119054, 7.05796877467521089360941920293, 7.11175493971575409100580667568, 7.14739672886643637708238059648, 7.43872689938246279510272419550, 7.84174519173191008412418554691, 8.071886781872829451147353063555, 8.382235023745211916485948384476
