| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s + 9-s − 2·10-s + 16-s − 2·17-s − 18-s + 2·20-s + 3·25-s + 6·29-s − 32-s + 2·34-s + 36-s + 4·37-s − 2·40-s − 12·41-s + 2·45-s − 6·49-s − 3·50-s + 4·53-s − 6·58-s − 8·61-s + 64-s − 2·68-s − 72-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.447·20-s + 3/5·25-s + 1.11·29-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.657·37-s − 0.316·40-s − 1.87·41-s + 0.298·45-s − 6/7·49-s − 0.424·50-s + 0.549·53-s − 0.787·58-s − 1.02·61-s + 1/8·64-s − 0.242·68-s − 0.117·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.987361795024339795662283769559, −7.32833413705432044347135260529, −6.85284220282740036840586761339, −6.55993095622274598408211011881, −6.26867307158483335186010775254, −5.56524513721541885838387047244, −5.26242109390800369165970363594, −4.68849314599593003136502163918, −4.16975831223379967729511728107, −3.55674599613018986821535090116, −2.68564362891618380770847463666, −2.62503701799559570764562418783, −1.58454257746921112913098418787, −1.31780003446384046459006095770, 0,
1.31780003446384046459006095770, 1.58454257746921112913098418787, 2.62503701799559570764562418783, 2.68564362891618380770847463666, 3.55674599613018986821535090116, 4.16975831223379967729511728107, 4.68849314599593003136502163918, 5.26242109390800369165970363594, 5.56524513721541885838387047244, 6.26867307158483335186010775254, 6.55993095622274598408211011881, 6.85284220282740036840586761339, 7.32833413705432044347135260529, 7.987361795024339795662283769559
