| L(s) = 1 | − 3·4-s + 3·9-s + 5·16-s − 10·19-s − 5·25-s − 14·29-s + 8·31-s − 9·36-s + 12·41-s + 2·49-s − 4·59-s − 16·61-s − 3·64-s + 2·71-s + 30·76-s − 2·79-s − 28·89-s + 15·100-s − 4·101-s + 32·109-s + 42·116-s − 22·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 9-s + 5/4·16-s − 2.29·19-s − 25-s − 2.59·29-s + 1.43·31-s − 3/2·36-s + 1.87·41-s + 2/7·49-s − 0.520·59-s − 2.04·61-s − 3/8·64-s + 0.237·71-s + 3.44·76-s − 0.225·79-s − 2.96·89-s + 3/2·100-s − 0.398·101-s + 3.06·109-s + 3.89·116-s − 2·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.368434051092139293667530691345, −9.264238826193943461669666199356, −8.637582098741098828567028517753, −8.166673073130889036064226586062, −7.58868145971317679966045525982, −7.20736947999270954805514319970, −6.18933898799275030960445851154, −6.04347889941024738587947019148, −5.17650250749336361526468916904, −4.50862835068296220400083952922, −4.11243849897383015215651683874, −3.81730422950962634095021138075, −2.54734060183809652565882169046, −1.57201196455560280184855777542, 0,
1.57201196455560280184855777542, 2.54734060183809652565882169046, 3.81730422950962634095021138075, 4.11243849897383015215651683874, 4.50862835068296220400083952922, 5.17650250749336361526468916904, 6.04347889941024738587947019148, 6.18933898799275030960445851154, 7.20736947999270954805514319970, 7.58868145971317679966045525982, 8.166673073130889036064226586062, 8.637582098741098828567028517753, 9.264238826193943461669666199356, 9.368434051092139293667530691345
