| L(s) = 1 | − 2-s + 4-s + 3·5-s − 2·7-s − 8-s − 9-s − 3·10-s + 2·14-s + 16-s + 18-s − 12·19-s + 3·20-s − 25-s − 2·28-s − 32-s − 6·35-s − 36-s + 8·37-s + 12·38-s − 3·40-s + 10·43-s − 3·45-s − 3·49-s + 50-s − 18·53-s + 2·56-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 1/3·9-s − 0.948·10-s + 0.534·14-s + 1/4·16-s + 0.235·18-s − 2.75·19-s + 0.670·20-s − 1/5·25-s − 0.377·28-s − 0.176·32-s − 1.01·35-s − 1/6·36-s + 1.31·37-s + 1.94·38-s − 0.474·40-s + 1.52·43-s − 0.447·45-s − 3/7·49-s + 0.141·50-s − 2.47·53-s + 0.267·56-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−9.087669628565527111185827071291, −8.491247215710411278276068611247, −7.966104914760167538449895424440, −7.65356310815664646861856861959, −6.65325604725093162981386703795, −6.48557995942947770405255122311, −6.08735840672394645070625105608, −5.73317648133547067089784351479, −4.91287437078239474171841185180, −4.24891565973373618487683710828, −3.60741161803455535874803927506, −2.58653518000747800397667958675, −2.34806771902777962339073820885, −1.51052207235870570546963347004, 0,
1.51052207235870570546963347004, 2.34806771902777962339073820885, 2.58653518000747800397667958675, 3.60741161803455535874803927506, 4.24891565973373618487683710828, 4.91287437078239474171841185180, 5.73317648133547067089784351479, 6.08735840672394645070625105608, 6.48557995942947770405255122311, 6.65325604725093162981386703795, 7.65356310815664646861856861959, 7.966104914760167538449895424440, 8.491247215710411278276068611247, 9.087669628565527111185827071291
