| L(s) = 1 | − 2·3-s − 4·4-s − 7-s + 9-s + 8·12-s + 3·13-s + 12·16-s + 7·19-s + 2·21-s − 25-s + 4·27-s + 4·28-s + 7·31-s − 4·36-s + 37-s − 6·39-s − 11·43-s − 24·48-s − 6·49-s − 12·52-s − 14·57-s + 61-s − 63-s − 32·64-s − 8·67-s − 8·73-s + 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s − 0.377·7-s + 1/3·9-s + 2.30·12-s + 0.832·13-s + 3·16-s + 1.60·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.755·28-s + 1.25·31-s − 2/3·36-s + 0.164·37-s − 0.960·39-s − 1.67·43-s − 3.46·48-s − 6/7·49-s − 1.66·52-s − 1.85·57-s + 0.128·61-s − 0.125·63-s − 4·64-s − 0.977·67-s − 0.936·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.995956220199515855850214356570, −7.71678866995883048306278166782, −6.95779371821822371322674608427, −6.50063985168643242611895997618, −5.94803167618980401203216519528, −5.72455979410018820898161607369, −5.14262261328787217108002739936, −4.82619579909272476409917933021, −4.49808452816436727701249466496, −3.69929538691913178788679023992, −3.43369672756294359861438708414, −2.83459074836064732049663532891, −1.41000832520587129717817504976, −0.877059684597282960479752388310, 0,
0.877059684597282960479752388310, 1.41000832520587129717817504976, 2.83459074836064732049663532891, 3.43369672756294359861438708414, 3.69929538691913178788679023992, 4.49808452816436727701249466496, 4.82619579909272476409917933021, 5.14262261328787217108002739936, 5.72455979410018820898161607369, 5.94803167618980401203216519528, 6.50063985168643242611895997618, 6.95779371821822371322674608427, 7.71678866995883048306278166782, 7.995956220199515855850214356570
