| L(s) = 1 | + 4·5-s + 2·7-s − 5·11-s + 2·13-s + 6·17-s + 2·19-s − 6·23-s + 5·25-s − 2·29-s + 4·31-s + 8·35-s − 16·37-s + 41-s + 7·43-s + 2·47-s + 7·49-s + 8·53-s − 20·55-s + 5·59-s + 8·65-s + 13·67-s − 16·71-s + 6·73-s − 10·77-s − 8·79-s + 12·83-s + 24·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s − 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s + 25-s − 0.371·29-s + 0.718·31-s + 1.35·35-s − 2.63·37-s + 0.156·41-s + 1.06·43-s + 0.291·47-s + 49-s + 1.09·53-s − 2.69·55-s + 0.650·59-s + 0.992·65-s + 1.58·67-s − 1.89·71-s + 0.702·73-s − 1.13·77-s − 0.900·79-s + 1.31·83-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.134510591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.134510591\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.24185861159593440189934703232, −9.974327441703596988277695031827, −9.801646277950892102085266682325, −9.102610563332336113690114902861, −8.698766388991177365963747798666, −8.320054423904554738941647315520, −7.83474457077398193746064331862, −7.39986399255915770341516754837, −7.11144172247950034886613152900, −6.12842629786114437513787895620, −6.02609998472896830680936156415, −5.57803597310641078041293321193, −5.11736791432871418710974132017, −4.96121407634399342717620088964, −3.93885302369468318928805366049, −3.51514822791257221694532360845, −2.73209669887380187936347512834, −2.13478423987345218728264761860, −1.80553128962034589871986119610, −0.887882465925395623322711351056,
0.887882465925395623322711351056, 1.80553128962034589871986119610, 2.13478423987345218728264761860, 2.73209669887380187936347512834, 3.51514822791257221694532360845, 3.93885302369468318928805366049, 4.96121407634399342717620088964, 5.11736791432871418710974132017, 5.57803597310641078041293321193, 6.02609998472896830680936156415, 6.12842629786114437513787895620, 7.11144172247950034886613152900, 7.39986399255915770341516754837, 7.83474457077398193746064331862, 8.320054423904554738941647315520, 8.698766388991177365963747798666, 9.102610563332336113690114902861, 9.801646277950892102085266682325, 9.974327441703596988277695031827, 10.24185861159593440189934703232
