| L(s) = 1 | + 2-s + 4-s + 8-s − 2·9-s − 4·11-s + 16-s − 2·18-s − 4·22-s + 4·29-s + 32-s − 2·36-s + 4·37-s + 4·43-s − 4·44-s − 7·49-s + 4·53-s + 4·58-s + 64-s − 4·67-s − 8·71-s − 2·72-s + 4·74-s − 16·79-s − 5·81-s + 4·86-s − 4·88-s − 7·98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s − 1.20·11-s + 1/4·16-s − 0.471·18-s − 0.852·22-s + 0.742·29-s + 0.176·32-s − 1/3·36-s + 0.657·37-s + 0.609·43-s − 0.603·44-s − 49-s + 0.549·53-s + 0.525·58-s + 1/8·64-s − 0.488·67-s − 0.949·71-s − 0.235·72-s + 0.464·74-s − 1.80·79-s − 5/9·81-s + 0.431·86-s − 0.426·88-s − 0.707·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−7.55569045641471438920038155756, −7.17461334550886289128087211438, −6.61298354675060689579344880077, −6.10863547183009214746843756581, −5.87696779022679300714086007088, −5.33700945213868632508927358837, −5.02060970334481432131409227407, −4.44028763037447032226633609883, −4.15404398667151229102648642656, −3.33333867064309230245759387291, −2.98021033161439585801371244380, −2.55962356589419839216055139508, −1.97649318503674693835060935633, −1.08762125366550256956802223029, 0,
1.08762125366550256956802223029, 1.97649318503674693835060935633, 2.55962356589419839216055139508, 2.98021033161439585801371244380, 3.33333867064309230245759387291, 4.15404398667151229102648642656, 4.44028763037447032226633609883, 5.02060970334481432131409227407, 5.33700945213868632508927358837, 5.87696779022679300714086007088, 6.10863547183009214746843756581, 6.61298354675060689579344880077, 7.17461334550886289128087211438, 7.55569045641471438920038155756
