| L(s) = 1 | − 10·7-s − 68·13-s + 128·19-s + 1.16e3·25-s + 1.39e3·31-s − 1.49e3·37-s − 5.23e3·43-s − 4.72e3·49-s + 1.28e4·61-s + 1.04e4·67-s − 9.03e3·73-s − 1.50e4·79-s + 680·91-s + 2.11e4·97-s + 1.16e4·103-s − 1.00e4·109-s + 1.55e4·121-s + 127-s + 131-s − 1.28e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.204·7-s − 0.402·13-s + 0.354·19-s + 1.87·25-s + 1.45·31-s − 1.09·37-s − 2.83·43-s − 1.96·49-s + 3.44·61-s + 2.32·67-s − 1.69·73-s − 2.40·79-s + 0.0821·91-s + 2.24·97-s + 1.09·103-s − 0.845·109-s + 1.06·121-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.0723·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.100162158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.100162158\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - 1169 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15593 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 34 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 35458 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 64 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 185138 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 286718 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 697 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 748 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5183666 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2618 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2758046 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14633921 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9567874 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6404 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5218 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7658462 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4519 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7502 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 64875281 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 46736606 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10571 T + p^{4} T^{2} )^{2} \) |
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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
−10.68918115665637423283838966369, −10.07820378319382933309568587270, −10.01964312669637317859716684862, −9.596634121700847561802868319186, −8.734227492320803443803696480487, −8.519312219571044341831132872063, −8.232791560014739302147420563550, −7.45606025443198245414961035692, −6.88392018319206240868966792698, −6.72006375706422888546555563664, −6.18347007224074524320106045469, −5.31887974141267104679505997299, −5.02620518126748071656722574306, −4.61394359760580232699193384008, −3.74045434850525621541896381032, −3.22514507476873214906039032901, −2.72800916874921561164442697910, −1.92062792018800602581848058174, −1.17940573978854530401552574712, −0.42989797161675293592963957028,
0.42989797161675293592963957028, 1.17940573978854530401552574712, 1.92062792018800602581848058174, 2.72800916874921561164442697910, 3.22514507476873214906039032901, 3.74045434850525621541896381032, 4.61394359760580232699193384008, 5.02620518126748071656722574306, 5.31887974141267104679505997299, 6.18347007224074524320106045469, 6.72006375706422888546555563664, 6.88392018319206240868966792698, 7.45606025443198245414961035692, 8.232791560014739302147420563550, 8.519312219571044341831132872063, 8.734227492320803443803696480487, 9.596634121700847561802868319186, 10.01964312669637317859716684862, 10.07820378319382933309568587270, 10.68918115665637423283838966369
