| L(s) = 1 | + 3·2-s + 3·4-s − 3·8-s − 9-s − 5·13-s − 13·16-s − 3·18-s − 8·19-s − 2·25-s − 15·26-s − 15·32-s − 3·36-s − 24·38-s − 12·43-s + 11·47-s + 4·49-s − 6·50-s − 15·52-s + 2·53-s + 11·59-s + 3·64-s − 14·67-s + 3·72-s − 24·76-s + 81-s − 23·83-s − 36·86-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 1.06·8-s − 1/3·9-s − 1.38·13-s − 3.25·16-s − 0.707·18-s − 1.83·19-s − 2/5·25-s − 2.94·26-s − 2.65·32-s − 1/2·36-s − 3.89·38-s − 1.82·43-s + 1.60·47-s + 4/7·49-s − 0.848·50-s − 2.08·52-s + 0.274·53-s + 1.43·59-s + 3/8·64-s − 1.71·67-s + 0.353·72-s − 2.75·76-s + 1/9·81-s − 2.52·83-s − 3.88·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122247 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122247 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
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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
−9.025144944775428939002286488798, −8.773843597244740696175159409910, −8.290948210138193963633312039034, −7.46402363446303646060571916781, −7.01671897858910485196269626025, −6.30953213856316227918532816957, −6.01830592999699460699604207701, −5.41279393824614459315915873865, −4.92567296475491828188526676949, −4.51771381908223612733109376689, −3.99456218240355859377676727938, −3.46989651135416634877765729787, −2.65052241186921972201892367382, −2.20380482134170068687789437074, 0,
2.20380482134170068687789437074, 2.65052241186921972201892367382, 3.46989651135416634877765729787, 3.99456218240355859377676727938, 4.51771381908223612733109376689, 4.92567296475491828188526676949, 5.41279393824614459315915873865, 6.01830592999699460699604207701, 6.30953213856316227918532816957, 7.01671897858910485196269626025, 7.46402363446303646060571916781, 8.290948210138193963633312039034, 8.773843597244740696175159409910, 9.025144944775428939002286488798
