| L(s) = 1 | − 3-s − 3·4-s − 4·7-s + 9-s + 3·12-s − 8·13-s + 5·16-s − 2·19-s + 4·21-s + 25-s − 27-s + 12·28-s − 3·36-s + 8·39-s − 20·43-s − 5·48-s − 2·49-s + 24·52-s + 2·57-s + 4·61-s − 4·63-s − 3·64-s − 32·67-s − 4·73-s − 75-s + 6·76-s − 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s + 0.866·12-s − 2.21·13-s + 5/4·16-s − 0.458·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 2.26·28-s − 1/2·36-s + 1.28·39-s − 3.04·43-s − 0.721·48-s − 2/7·49-s + 3.32·52-s + 0.264·57-s + 0.512·61-s − 0.503·63-s − 3/8·64-s − 3.90·67-s − 0.468·73-s − 0.115·75-s + 0.688·76-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.450551892023889738347528214751, −8.255821268467971823002210114232, −7.28034875915901512150776673175, −7.20655519160802512316474001684, −6.50028762370085906668100412134, −6.11724584120824685520629468061, −5.26580762429388022199308033556, −5.15467699824668629469430799137, −4.39966598748280163950451872807, −4.19187417186236300755390321977, −3.13810729229125509539196623139, −2.93269055521134332794719136670, −1.68584050985555553185267943448, 0, 0,
1.68584050985555553185267943448, 2.93269055521134332794719136670, 3.13810729229125509539196623139, 4.19187417186236300755390321977, 4.39966598748280163950451872807, 5.15467699824668629469430799137, 5.26580762429388022199308033556, 6.11724584120824685520629468061, 6.50028762370085906668100412134, 7.20655519160802512316474001684, 7.28034875915901512150776673175, 8.255821268467971823002210114232, 8.450551892023889738347528214751
