| L(s) = 1 | + 4-s + 2·5-s − 4·7-s − 5·9-s − 2·13-s + 16-s + 2·20-s + 8·23-s − 7·25-s − 4·28-s − 29-s − 8·35-s − 5·36-s − 10·45-s − 2·49-s − 2·52-s − 22·53-s + 20·63-s + 64-s − 4·65-s − 24·67-s + 4·71-s + 2·80-s + 16·81-s + 8·83-s + 8·91-s + 8·92-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s − 1.51·7-s − 5/3·9-s − 0.554·13-s + 1/4·16-s + 0.447·20-s + 1.66·23-s − 7/5·25-s − 0.755·28-s − 0.185·29-s − 1.35·35-s − 5/6·36-s − 1.49·45-s − 2/7·49-s − 0.277·52-s − 3.02·53-s + 2.51·63-s + 1/8·64-s − 0.496·65-s − 2.93·67-s + 0.474·71-s + 0.223·80-s + 16/9·81-s + 0.878·83-s + 0.838·91-s + 0.834·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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
−9.467218583128971546085930879240, −9.024969058339305348445790513847, −8.471610187154116739894076984158, −7.71436185563727090632996745083, −7.40876305943440204027892451644, −6.55316050758606037288366864617, −6.11658999858863662469236119804, −6.08976797776307046150097887252, −5.25136790623537531107690201816, −4.78875549833122639002516063824, −3.58875335460876869693637888804, −3.07071881135695746551942332895, −2.69470363093768135991873388448, −1.76909934994324200300399553047, 0,
1.76909934994324200300399553047, 2.69470363093768135991873388448, 3.07071881135695746551942332895, 3.58875335460876869693637888804, 4.78875549833122639002516063824, 5.25136790623537531107690201816, 6.08976797776307046150097887252, 6.11658999858863662469236119804, 6.55316050758606037288366864617, 7.40876305943440204027892451644, 7.71436185563727090632996745083, 8.471610187154116739894076984158, 9.024969058339305348445790513847, 9.467218583128971546085930879240
