| L(s) = 1 | − 3·3-s − 3·7-s + 6·9-s − 6·11-s + 6·13-s − 6·19-s + 9·21-s + 4·23-s − 10·27-s + 2·29-s − 2·31-s + 18·33-s + 4·37-s − 18·39-s + 2·41-s + 4·43-s + 8·47-s + 6·49-s − 14·53-s + 18·57-s − 16·59-s − 6·61-s − 18·63-s + 8·67-s − 12·69-s − 6·71-s + 18·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 2·9-s − 1.80·11-s + 1.66·13-s − 1.37·19-s + 1.96·21-s + 0.834·23-s − 1.92·27-s + 0.371·29-s − 0.359·31-s + 3.13·33-s + 0.657·37-s − 2.88·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 6/7·49-s − 1.92·53-s + 2.38·57-s − 2.08·59-s − 0.768·61-s − 2.26·63-s + 0.977·67-s − 1.44·69-s − 0.712·71-s + 2.10·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 168 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T - 15 T^{2} + 488 T^{3} - 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 169 T^{2} - 944 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 311 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.37782437368795903215499253676, −6.68081655476558727680969104131, −6.60078975998565556985431380295, −6.58338176705409905696685364759, −6.15560448225360899864381147422, −6.15069586717818494955415685431, −6.05539313801456562224250870081, −5.54543617506253666716047371260, −5.40781943338052381483785215728, −5.27388068402682408063975141345, −4.83325597672392421574458417536, −4.75416492917165427181348388054, −4.55507526906904770777796147130, −4.07429432506124573827791717692, −3.92433495228808534049156916475, −3.80460676777693763905331581314, −3.25702646945647185951599478160, −3.05747886626453827216832863808, −3.02324142082818023663867859928, −2.26160975308956713656624716383, −2.25946210662107794823947702155, −2.11742007348143788565574789827, −1.16269552632115592592377445364, −1.10037261093260786587223543658, −1.09342486855762168527640977762, 0, 0, 0,
1.09342486855762168527640977762, 1.10037261093260786587223543658, 1.16269552632115592592377445364, 2.11742007348143788565574789827, 2.25946210662107794823947702155, 2.26160975308956713656624716383, 3.02324142082818023663867859928, 3.05747886626453827216832863808, 3.25702646945647185951599478160, 3.80460676777693763905331581314, 3.92433495228808534049156916475, 4.07429432506124573827791717692, 4.55507526906904770777796147130, 4.75416492917165427181348388054, 4.83325597672392421574458417536, 5.27388068402682408063975141345, 5.40781943338052381483785215728, 5.54543617506253666716047371260, 6.05539313801456562224250870081, 6.15069586717818494955415685431, 6.15560448225360899864381147422, 6.58338176705409905696685364759, 6.60078975998565556985431380295, 6.68081655476558727680969104131, 7.37782437368795903215499253676
