| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 5-s + 9·6-s − 5·7-s − 3·8-s + 6·9-s + 3·10-s − 6·11-s − 12·12-s + 15·14-s + 3·15-s + 3·16-s − 6·17-s − 18·18-s − 12·19-s − 4·20-s + 15·21-s + 18·22-s + 3·23-s + 9·24-s − 9·27-s − 20·28-s − 9·30-s − 6·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 0.447·5-s + 3.67·6-s − 1.88·7-s − 1.06·8-s + 2·9-s + 0.948·10-s − 1.80·11-s − 3.46·12-s + 4.00·14-s + 0.774·15-s + 3/4·16-s − 1.45·17-s − 4.24·18-s − 2.75·19-s − 0.894·20-s + 3.27·21-s + 3.83·22-s + 0.625·23-s + 1.83·24-s − 1.73·27-s − 3.77·28-s − 1.64·30-s − 1.07·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−12.86030469335877291228631811314, −12.81278060750407138636131065364, −12.70460821430385747282035445610, −11.68384543798572901545518948867, −10.88671961733396961010145934687, −10.79937734987930125606157001943, −10.20478907525436304041427489254, −10.07409954279382640044032862993, −8.983541384324832939876145785173, −8.974408970577501580761814618917, −8.102770939601159033698036488876, −7.43357468040600120656989299546, −6.59014411661458313488026358895, −6.57616908830316756187719347272, −5.65540329908988067541756759561, −4.82403773402819725151125010044, −3.81214811365540460137061738021, −2.37166676432777566319260691004, 0, 0,
2.37166676432777566319260691004, 3.81214811365540460137061738021, 4.82403773402819725151125010044, 5.65540329908988067541756759561, 6.57616908830316756187719347272, 6.59014411661458313488026358895, 7.43357468040600120656989299546, 8.102770939601159033698036488876, 8.974408970577501580761814618917, 8.983541384324832939876145785173, 10.07409954279382640044032862993, 10.20478907525436304041427489254, 10.79937734987930125606157001943, 10.88671961733396961010145934687, 11.68384543798572901545518948867, 12.70460821430385747282035445610, 12.81278060750407138636131065364, 12.86030469335877291228631811314
