| L(s) = 1 | + 4·3-s + 7-s + 6·9-s − 4·19-s + 4·21-s + 6·25-s − 4·27-s + 4·29-s + 8·31-s − 12·37-s − 8·47-s + 49-s − 20·53-s − 16·57-s + 12·59-s + 6·63-s + 24·75-s − 37·81-s + 12·83-s + 16·87-s + 32·93-s − 24·103-s + 20·109-s − 48·111-s + 12·113-s − 22·121-s + 127-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.377·7-s + 2·9-s − 0.917·19-s + 0.872·21-s + 6/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.97·37-s − 1.16·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s + 1.56·59-s + 0.755·63-s + 2.77·75-s − 4.11·81-s + 1.31·83-s + 1.71·87-s + 3.31·93-s − 2.36·103-s + 1.91·109-s − 4.55·111-s + 1.12·113-s − 2·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.717535060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.717535060\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.835763078329112662938551691511, −9.676028605922657643630885041741, −8.928329697232844766585907221767, −8.589598276995096865608741769842, −8.188057214509100026281306554311, −8.037894346653340821155380595819, −7.14438434722230698367292292955, −6.69299348503088401254902880103, −5.96146004102990052123544748972, −4.99083605129769608898331014735, −4.50185552371175935345657331179, −3.56791835941750047835108384805, −3.14137792992390816749783057891, −2.48337724162238959661867349800, −1.73163501201493931998851829334,
1.73163501201493931998851829334, 2.48337724162238959661867349800, 3.14137792992390816749783057891, 3.56791835941750047835108384805, 4.50185552371175935345657331179, 4.99083605129769608898331014735, 5.96146004102990052123544748972, 6.69299348503088401254902880103, 7.14438434722230698367292292955, 8.037894346653340821155380595819, 8.188057214509100026281306554311, 8.589598276995096865608741769842, 8.928329697232844766585907221767, 9.676028605922657643630885041741, 9.835763078329112662938551691511
