| L(s) = 1 | − 3-s − 4·5-s + 9-s + 4·15-s − 8·19-s − 16·23-s + 2·25-s − 27-s + 12·29-s + 8·43-s − 4·45-s − 14·49-s − 4·53-s + 8·57-s − 8·67-s + 16·69-s + 16·71-s + 20·73-s − 2·75-s + 81-s − 12·87-s + 32·95-s + 4·97-s − 36·101-s + 64·115-s − 6·121-s + 28·125-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s − 1.83·19-s − 3.33·23-s + 2/5·25-s − 0.192·27-s + 2.22·29-s + 1.21·43-s − 0.596·45-s − 2·49-s − 0.549·53-s + 1.05·57-s − 0.977·67-s + 1.92·69-s + 1.89·71-s + 2.34·73-s − 0.230·75-s + 1/9·81-s − 1.28·87-s + 3.28·95-s + 0.406·97-s − 3.58·101-s + 5.96·115-s − 0.545·121-s + 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98603906202358244044014470993, −10.54464146795679939621303070304, −9.964671915727173343349228861429, −9.381546175761683500969667585237, −8.240414794588996598510176234846, −8.098990694093691505068092710868, −7.87582021380809768630528089037, −6.70029239224408692327848930668, −6.42897107072744719896577758454, −5.67110067949068956351494574607, −4.48990443653583983179988972356, −4.25303028692796488061253338187, −3.59090248724659668634540115790, −2.18788103243241609268384022128, 0,
2.18788103243241609268384022128, 3.59090248724659668634540115790, 4.25303028692796488061253338187, 4.48990443653583983179988972356, 5.67110067949068956351494574607, 6.42897107072744719896577758454, 6.70029239224408692327848930668, 7.87582021380809768630528089037, 8.098990694093691505068092710868, 8.240414794588996598510176234846, 9.381546175761683500969667585237, 9.964671915727173343349228861429, 10.54464146795679939621303070304, 10.98603906202358244044014470993
