| L(s) = 1 | − 3·2-s + 6·4-s + 3·5-s − 3·7-s − 10·8-s − 9·10-s − 9·13-s + 9·14-s + 15·16-s + 6·17-s − 3·19-s + 18·20-s + 6·23-s − 6·25-s + 27·26-s − 18·28-s + 3·29-s − 9·31-s − 21·32-s − 18·34-s − 9·35-s − 6·37-s + 9·38-s − 30·40-s + 6·41-s − 18·46-s + 15·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.34·5-s − 1.13·7-s − 3.53·8-s − 2.84·10-s − 2.49·13-s + 2.40·14-s + 15/4·16-s + 1.45·17-s − 0.688·19-s + 4.02·20-s + 1.25·23-s − 6/5·25-s + 5.29·26-s − 3.40·28-s + 0.557·29-s − 1.61·31-s − 3.71·32-s − 3.08·34-s − 1.52·35-s − 0.986·37-s + 1.45·38-s − 4.74·40-s + 0.937·41-s − 2.65·46-s + 2.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 223^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 223^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 223 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 29 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 41 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 24 T^{2} + 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 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 | $A_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 131 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 111 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 225 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 27 T^{2} + 59 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 9 T + 81 T^{2} + 379 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 12 T^{2} - 203 T^{3} + 12 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 333 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 102 T^{2} + 27 T^{3} + 102 p T^{4} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 15 T + 123 T^{2} - 781 T^{3} + 123 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 9 T + 57 T^{2} - 145 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 711 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 237 T^{2} + 1833 T^{3} + 237 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 653 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 73 | $A_4\times C_2$ | \( 1 + 12 T + 246 T^{2} + 1769 T^{3} + 246 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 216 T^{2} - 37 T^{3} + 216 p T^{4} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 819 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 3 T + 141 T^{2} + 477 T^{3} + 141 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 27 T + 507 T^{2} + 5697 T^{3} + 507 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.82932643748712263761104073303, −7.59094137148807213867702744360, −7.42362669568095557824902335656, −7.27301114565281815207124495406, −6.80876565515720870745823684546, −6.73669820501435009875328852977, −6.69950395528874181409006074460, −6.04266341002049480781776292703, −5.90811604316174739294307818588, −5.87232654099195175732320365904, −5.28331182001181423615977909443, −5.27561896728964916097160413630, −5.23869489967962269298029900678, −4.40675880297888010775039975094, −4.26565188646406035700807437304, −4.00674630171156611346783483968, −3.36150843786468517880454984592, −3.23457573479549653298435554648, −2.92921110256464231050264437429, −2.44982007721010542055685786850, −2.42590670274807802506171257024, −2.27014236200276274535337146358, −1.46598680503735829090595980918, −1.42088029083976353758203242934, −1.21694550193589941713477804393, 0, 0, 0,
1.21694550193589941713477804393, 1.42088029083976353758203242934, 1.46598680503735829090595980918, 2.27014236200276274535337146358, 2.42590670274807802506171257024, 2.44982007721010542055685786850, 2.92921110256464231050264437429, 3.23457573479549653298435554648, 3.36150843786468517880454984592, 4.00674630171156611346783483968, 4.26565188646406035700807437304, 4.40675880297888010775039975094, 5.23869489967962269298029900678, 5.27561896728964916097160413630, 5.28331182001181423615977909443, 5.87232654099195175732320365904, 5.90811604316174739294307818588, 6.04266341002049480781776292703, 6.69950395528874181409006074460, 6.73669820501435009875328852977, 6.80876565515720870745823684546, 7.27301114565281815207124495406, 7.42362669568095557824902335656, 7.59094137148807213867702744360, 7.82932643748712263761104073303
