| L(s) = 1 | + 3-s − 4-s − 3·9-s − 12-s + 13-s + 16-s − 9·23-s + 25-s − 4·27-s + 6·29-s + 3·36-s + 39-s − 11·43-s + 48-s + 8·49-s − 52-s − 12·53-s − 4·61-s − 64-s − 9·69-s + 75-s − 16·79-s + 2·81-s + 6·87-s + 9·92-s − 100-s − 12·101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 9-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 1.87·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1/2·36-s + 0.160·39-s − 1.67·43-s + 0.144·48-s + 8/7·49-s − 0.138·52-s − 1.64·53-s − 0.512·61-s − 1/8·64-s − 1.08·69-s + 0.115·75-s − 1.80·79-s + 2/9·81-s + 0.643·87-s + 0.938·92-s − 0.0999·100-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
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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
−8.620056646137391415207872838299, −8.501889230987984583298775213736, −8.118003073494782356055322607866, −7.63014388209405679892092886615, −6.94533953548344418914297995747, −6.36260427009866872761395967849, −5.90693824280976848006739660881, −5.45881026226214572601893735585, −4.74922329955838983771360746062, −4.25278328243955123386913706151, −3.58186748312298307525934334737, −3.04115937105794354267619916574, −2.41435960477882200607881953721, −1.49051095708223809650216443346, 0,
1.49051095708223809650216443346, 2.41435960477882200607881953721, 3.04115937105794354267619916574, 3.58186748312298307525934334737, 4.25278328243955123386913706151, 4.74922329955838983771360746062, 5.45881026226214572601893735585, 5.90693824280976848006739660881, 6.36260427009866872761395967849, 6.94533953548344418914297995747, 7.63014388209405679892092886615, 8.118003073494782356055322607866, 8.501889230987984583298775213736, 8.620056646137391415207872838299
