| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s + 9-s − 3·10-s − 8·13-s + 16-s − 8·17-s − 18-s + 3·20-s + 2·25-s + 8·26-s − 6·29-s − 32-s + 8·34-s + 36-s − 8·37-s − 3·40-s + 8·41-s + 3·45-s + 2·49-s − 2·50-s − 8·52-s − 8·53-s + 6·58-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 2.21·13-s + 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.670·20-s + 2/5·25-s + 1.56·26-s − 1.11·29-s − 0.176·32-s + 1.37·34-s + 1/6·36-s − 1.31·37-s − 0.474·40-s + 1.24·41-s + 0.447·45-s + 2/7·49-s − 0.282·50-s − 1.10·52-s − 1.09·53-s + 0.787·58-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162720 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162720 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.138894850477068878477468769093, −8.751486743561878434120276019094, −7.963444567785560505390409494116, −7.54557904609640795741225449255, −7.01033250603076141516629381178, −6.67762608780319520787347587695, −6.13957422114397346125869898964, −5.47171213283511215520955543794, −5.03835870503198278921134994510, −4.47856536609905068324925128611, −3.69664640174762450133833669785, −2.50566399450932697554858398515, −2.35142789083797506035334899140, −1.65286199981960292429690052955, 0,
1.65286199981960292429690052955, 2.35142789083797506035334899140, 2.50566399450932697554858398515, 3.69664640174762450133833669785, 4.47856536609905068324925128611, 5.03835870503198278921134994510, 5.47171213283511215520955543794, 6.13957422114397346125869898964, 6.67762608780319520787347587695, 7.01033250603076141516629381178, 7.54557904609640795741225449255, 7.963444567785560505390409494116, 8.751486743561878434120276019094, 9.138894850477068878477468769093
