| L(s) = 1 | − 2·3-s + 2·5-s − 3·9-s + 11-s − 4·15-s − 4·16-s − 2·23-s − 7·25-s + 14·27-s + 14·31-s − 2·33-s + 6·37-s − 6·45-s + 16·47-s + 8·48-s − 10·49-s − 12·53-s + 2·55-s + 10·59-s − 14·67-s + 4·69-s − 6·71-s + 14·75-s − 8·80-s − 4·81-s + 30·89-s − 28·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 9-s + 0.301·11-s − 1.03·15-s − 16-s − 0.417·23-s − 7/5·25-s + 2.69·27-s + 2.51·31-s − 0.348·33-s + 0.986·37-s − 0.894·45-s + 2.33·47-s + 1.15·48-s − 1.42·49-s − 1.64·53-s + 0.269·55-s + 1.30·59-s − 1.71·67-s + 0.481·69-s − 0.712·71-s + 1.61·75-s − 0.894·80-s − 4/9·81-s + 3.17·89-s − 2.90·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4466091259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4466091259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−13.62205943269834176065110376300, −13.56863905712999451342489435843, −12.33713976074744055894961281844, −11.81631087306955963420628700213, −11.45125861034521058353374083886, −10.76517194615392112496312976161, −10.03550909718107888433464868208, −9.366222803316740054530803267641, −8.603539619290756001226038948684, −7.78789607043645765017922373874, −6.36261389471308870138602900888, −6.23870064789214498217427813253, −5.41175071356617719099292784896, −4.44574490854809981150981932393, −2.63044898935838963010520851422,
2.63044898935838963010520851422, 4.44574490854809981150981932393, 5.41175071356617719099292784896, 6.23870064789214498217427813253, 6.36261389471308870138602900888, 7.78789607043645765017922373874, 8.603539619290756001226038948684, 9.366222803316740054530803267641, 10.03550909718107888433464868208, 10.76517194615392112496312976161, 11.45125861034521058353374083886, 11.81631087306955963420628700213, 12.33713976074744055894961281844, 13.56863905712999451342489435843, 13.62205943269834176065110376300
