| L(s) = 1 | + 3-s − 3·5-s − 7-s − 11-s − 8·13-s − 3·15-s − 4·17-s − 21-s + 8·23-s + 5·25-s − 27-s − 14·29-s + 11·31-s − 33-s + 3·35-s − 4·37-s − 8·39-s − 8·41-s + 4·43-s − 2·47-s − 6·49-s − 4·51-s + 11·53-s + 3·55-s + 7·59-s − 10·61-s + 24·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s − 0.301·11-s − 2.21·13-s − 0.774·15-s − 0.970·17-s − 0.218·21-s + 1.66·23-s + 25-s − 0.192·27-s − 2.59·29-s + 1.97·31-s − 0.174·33-s + 0.507·35-s − 0.657·37-s − 1.28·39-s − 1.24·41-s + 0.609·43-s − 0.291·47-s − 6/7·49-s − 0.560·51-s + 1.51·53-s + 0.404·55-s + 0.911·59-s − 1.28·61-s + 2.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6687055760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6687055760\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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
−10.82018412178712492693989784384, −10.16770161126474226271911820346, −9.882724046586111032315442386007, −9.243528528350979765049350673891, −9.143411649362586696831363370051, −8.459368790109108018929716642288, −8.142376629791072338346520784747, −7.60918841440392638251361485796, −7.30564475122720038630997502135, −6.79446746760302646877784681703, −6.69277756068589216549189124943, −5.41150339553893573471232803649, −5.39845658638549212085684967099, −4.44874214861656310617675544730, −4.43565797025948028247464948349, −3.56959191540855915071565191122, −3.03274043102579757558441036901, −2.61388335700796175884529320249, −1.88000897355429537210384023582, −0.40301410796786540700582645278,
0.40301410796786540700582645278, 1.88000897355429537210384023582, 2.61388335700796175884529320249, 3.03274043102579757558441036901, 3.56959191540855915071565191122, 4.43565797025948028247464948349, 4.44874214861656310617675544730, 5.39845658638549212085684967099, 5.41150339553893573471232803649, 6.69277756068589216549189124943, 6.79446746760302646877784681703, 7.30564475122720038630997502135, 7.60918841440392638251361485796, 8.142376629791072338346520784747, 8.459368790109108018929716642288, 9.143411649362586696831363370051, 9.243528528350979765049350673891, 9.882724046586111032315442386007, 10.16770161126474226271911820346, 10.82018412178712492693989784384
