| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 5·7-s − 10·8-s − 6·9-s + 4·11-s − 6·12-s + 15·14-s + 15·16-s − 10·17-s + 18·18-s + 8·19-s + 5·21-s − 12·22-s + 3·23-s + 10·24-s + 8·27-s − 30·28-s + 3·29-s + 8·31-s − 21·32-s − 4·33-s + 30·34-s − 36·36-s − 8·37-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 3·4-s + 1.22·6-s − 1.88·7-s − 3.53·8-s − 2·9-s + 1.20·11-s − 1.73·12-s + 4.00·14-s + 15/4·16-s − 2.42·17-s + 4.24·18-s + 1.83·19-s + 1.09·21-s − 2.55·22-s + 0.625·23-s + 2.04·24-s + 1.53·27-s − 5.66·28-s + 0.557·29-s + 1.43·31-s − 3.71·32-s − 0.696·33-s + 5.14·34-s − 6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + T + 7 T^{2} + 5 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 5 T + 27 T^{2} + 71 T^{3} + 27 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 4 T + 29 T^{2} - 80 T^{3} + 29 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 10 T + 75 T^{2} + 348 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 8 T + 69 T^{2} - 296 T^{3} + 69 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + p T^{2} + T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 83 T^{2} - 175 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 8 T + 49 T^{2} - 152 T^{3} + 49 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 95 T^{2} + 528 T^{3} + 95 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 11 T + 49 T^{2} - 75 T^{3} + 49 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - T + 99 T^{2} - p T^{3} + 99 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 15 T + 167 T^{2} + 1339 T^{3} + 167 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 4 T + 43 T^{2} - 144 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 24 T + 341 T^{2} - 3176 T^{3} + 341 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 167 T^{2} + 1297 T^{3} + 167 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 5 T + 123 T^{2} + 839 T^{3} + 123 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 32 T + 545 T^{2} - 5656 T^{3} + 545 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 26 T + 379 T^{2} - 3692 T^{3} + 379 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 2 T + 117 T^{2} + 28 T^{3} + 117 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 95 T^{2} + 875 T^{3} + 95 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 17 T + 137 T^{2} + 829 T^{3} + 137 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 107 T^{2} - 52 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.20363806346287051443760628570, −6.75412431087061691975594237819, −6.72603794088675524044972166436, −6.61929976799317476721420305366, −6.42136723202936707001243152821, −6.31238119614408187286287266538, −6.20711506639497732205047362994, −5.66972864646414632082570494077, −5.40370370873342952750748604681, −5.38596293692668582750071792756, −4.96718115397100393994767249677, −4.66526508132643094861357676197, −4.57791887230525940069780036990, −3.74246537941684237003829277719, −3.69806730328256745835434080172, −3.63900489672802512318421220203, −3.23348364405094769535997179108, −2.82988841483674989892816608072, −2.77655724850383932714564337213, −2.42951573735860727804134298301, −2.31507886956792160807691217125, −1.87018751949396259551975712850, −1.25336435370360182006584308702, −1.01762206673394631168629824047, −0.893820130382505929193789809041, 0, 0, 0,
0.893820130382505929193789809041, 1.01762206673394631168629824047, 1.25336435370360182006584308702, 1.87018751949396259551975712850, 2.31507886956792160807691217125, 2.42951573735860727804134298301, 2.77655724850383932714564337213, 2.82988841483674989892816608072, 3.23348364405094769535997179108, 3.63900489672802512318421220203, 3.69806730328256745835434080172, 3.74246537941684237003829277719, 4.57791887230525940069780036990, 4.66526508132643094861357676197, 4.96718115397100393994767249677, 5.38596293692668582750071792756, 5.40370370873342952750748604681, 5.66972864646414632082570494077, 6.20711506639497732205047362994, 6.31238119614408187286287266538, 6.42136723202936707001243152821, 6.61929976799317476721420305366, 6.72603794088675524044972166436, 6.75412431087061691975594237819, 7.20363806346287051443760628570
