| L(s) = 1 | − 3-s − 4·4-s − 2·9-s + 4·12-s − 10·13-s + 12·16-s − 4·19-s + 25-s + 5·27-s + 8·31-s + 8·36-s + 4·37-s + 10·39-s − 20·43-s − 12·48-s + 40·52-s + 4·57-s − 16·61-s − 32·64-s − 8·67-s − 4·73-s − 75-s + 16·76-s − 2·79-s + 81-s − 8·93-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 2/3·9-s + 1.15·12-s − 2.77·13-s + 3·16-s − 0.917·19-s + 1/5·25-s + 0.962·27-s + 1.43·31-s + 4/3·36-s + 0.657·37-s + 1.60·39-s − 3.04·43-s − 1.73·48-s + 5.54·52-s + 0.529·57-s − 2.04·61-s − 4·64-s − 0.977·67-s − 0.468·73-s − 0.115·75-s + 1.83·76-s − 0.225·79-s + 1/9·81-s − 0.829·93-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−7.959921722640743022302973334932, −7.80156257853413862136981277007, −7.21456919347212808652226059968, −6.36641447795918613892295316962, −6.30078072165756181986797640838, −5.30121167828360660438204787375, −5.19374718417958536543626274426, −4.71225653823478357730435111839, −4.48390865811726709341858080699, −3.78858238344580500375502888439, −2.94585098531436589990293317610, −2.61198254943438177491615875407, −1.37785913473779166712390600334, 0, 0,
1.37785913473779166712390600334, 2.61198254943438177491615875407, 2.94585098531436589990293317610, 3.78858238344580500375502888439, 4.48390865811726709341858080699, 4.71225653823478357730435111839, 5.19374718417958536543626274426, 5.30121167828360660438204787375, 6.30078072165756181986797640838, 6.36641447795918613892295316962, 7.21456919347212808652226059968, 7.80156257853413862136981277007, 7.959921722640743022302973334932
