| L(s) = 1 | − 2·4-s − 6·7-s + 9-s + 4·16-s − 6·17-s − 9·23-s − 2·25-s + 12·28-s − 4·31-s − 2·36-s + 4·41-s + 14·49-s − 6·63-s − 8·64-s + 12·68-s + 20·71-s + 4·73-s − 6·79-s + 81-s + 22·89-s + 18·92-s + 4·100-s − 6·103-s − 24·112-s − 6·113-s + 36·119-s − 2·121-s + ⋯ |
| L(s) = 1 | − 4-s − 2.26·7-s + 1/3·9-s + 16-s − 1.45·17-s − 1.87·23-s − 2/5·25-s + 2.26·28-s − 0.718·31-s − 1/3·36-s + 0.624·41-s + 2·49-s − 0.755·63-s − 64-s + 1.45·68-s + 2.37·71-s + 0.468·73-s − 0.675·79-s + 1/9·81-s + 2.33·89-s + 1.87·92-s + 2/5·100-s − 0.591·103-s − 2.26·112-s − 0.564·113-s + 3.30·119-s − 0.181·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−10.80909341329221947706372331960, −10.21673943244944906076985283447, −9.793752964803145119627992179500, −9.361232910820385093867401680480, −9.009642922206400074693419433239, −8.216946230740007981339406310676, −7.60218311857630484763298250624, −6.72636138873287628872984077978, −6.32135824504299330317413229889, −5.75856720894753881050859753020, −4.79867555432336924848293414780, −3.89023431743170574330800787922, −3.60195423474801674863178389375, −2.37486694852754809943483341734, 0,
2.37486694852754809943483341734, 3.60195423474801674863178389375, 3.89023431743170574330800787922, 4.79867555432336924848293414780, 5.75856720894753881050859753020, 6.32135824504299330317413229889, 6.72636138873287628872984077978, 7.60218311857630484763298250624, 8.216946230740007981339406310676, 9.009642922206400074693419433239, 9.361232910820385093867401680480, 9.793752964803145119627992179500, 10.21673943244944906076985283447, 10.80909341329221947706372331960
