| L(s) = 1 | − 4·2-s + 8·4-s − 2·5-s + 4·7-s − 8·8-s + 8·10-s − 2·11-s − 13-s − 16·14-s − 4·16-s − 19-s − 16·20-s + 8·22-s + 5·25-s + 4·26-s + 32·28-s + 4·29-s + 18·31-s + 32·32-s − 8·35-s − 3·37-s + 4·38-s + 16·40-s − 10·41-s − 5·43-s − 16·44-s + 12·47-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 0.894·5-s + 1.51·7-s − 2.82·8-s + 2.52·10-s − 0.603·11-s − 0.277·13-s − 4.27·14-s − 16-s − 0.229·19-s − 3.57·20-s + 1.70·22-s + 25-s + 0.784·26-s + 6.04·28-s + 0.742·29-s + 3.23·31-s + 5.65·32-s − 1.35·35-s − 0.493·37-s + 0.648·38-s + 2.52·40-s − 1.56·41-s − 0.762·43-s − 2.41·44-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4096229316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4096229316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.54292814594785839698246632952, −10.46519035573580123206572661450, −10.00325602139122528685860517666, −9.866467448781055346531118807751, −8.811594547916569395891583823060, −8.710059991180256479448302466236, −8.420404754779266631736035884639, −8.116171427437992090811813248278, −7.67887654930571629220856306229, −7.35134898220702317134842785728, −6.63819730963432706768209365275, −6.62987911610860197772291751306, −5.14619099985374657233347218035, −5.08169137202028105035760988109, −4.38934061282071355395441397692, −3.81331394102401450974069270103, −2.40763931081668169837431901317, −2.35863913059867466894053961448, −1.12951251165074874562472386512, −0.72721348267346860203987142512,
0.72721348267346860203987142512, 1.12951251165074874562472386512, 2.35863913059867466894053961448, 2.40763931081668169837431901317, 3.81331394102401450974069270103, 4.38934061282071355395441397692, 5.08169137202028105035760988109, 5.14619099985374657233347218035, 6.62987911610860197772291751306, 6.63819730963432706768209365275, 7.35134898220702317134842785728, 7.67887654930571629220856306229, 8.116171427437992090811813248278, 8.420404754779266631736035884639, 8.710059991180256479448302466236, 8.811594547916569395891583823060, 9.866467448781055346531118807751, 10.00325602139122528685860517666, 10.46519035573580123206572661450, 10.54292814594785839698246632952
