| L(s) = 1 | − 4·4-s + 114·11-s + 16·16-s + 212·19-s − 420·29-s + 94·31-s + 12·41-s − 456·44-s − 155·49-s + 168·59-s + 112·61-s − 64·64-s − 720·71-s − 848·76-s + 320·79-s − 1.90e3·89-s + 726·101-s − 3.46e3·109-s + 1.68e3·116-s + 7.08e3·121-s − 376·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 3.12·11-s + 1/4·16-s + 2.55·19-s − 2.68·29-s + 0.544·31-s + 0.0457·41-s − 1.56·44-s − 0.451·49-s + 0.370·59-s + 0.235·61-s − 1/8·64-s − 1.20·71-s − 1.27·76-s + 0.455·79-s − 2.27·89-s + 0.715·101-s − 3.04·109-s + 1.34·116-s + 5.32·121-s − 0.272·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.663188618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.663188618\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 155 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 57 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4642 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5942 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 101302 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 111490 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 17030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291193 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 581362 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 565247 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 603349 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 954 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1788865 T^{2} + p^{6} T^{4} \) |
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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
−9.348488303049488144379054139012, −9.270509076169233883855258397314, −8.859655166998024647620826184323, −8.101128957763037180460031134665, −7.996650752885588448376518018471, −7.23924360213739128865384479912, −6.89542905473033720653348949008, −6.85635989287206300495576340343, −5.97878596534447707558817415092, −5.71620871702143470721776051255, −5.41247446872237432445056676571, −4.68949183184355313443627819426, −4.16816466372746517723579701517, −3.94062863475010034017582896462, −3.29276604729768912207671742924, −3.19727429921465651927532247640, −2.07737396755249181123910991963, −1.38370381395521553047802982760, −1.24973816061674014263088229712, −0.49037539081063761991163497712,
0.49037539081063761991163497712, 1.24973816061674014263088229712, 1.38370381395521553047802982760, 2.07737396755249181123910991963, 3.19727429921465651927532247640, 3.29276604729768912207671742924, 3.94062863475010034017582896462, 4.16816466372746517723579701517, 4.68949183184355313443627819426, 5.41247446872237432445056676571, 5.71620871702143470721776051255, 5.97878596534447707558817415092, 6.85635989287206300495576340343, 6.89542905473033720653348949008, 7.23924360213739128865384479912, 7.996650752885588448376518018471, 8.101128957763037180460031134665, 8.859655166998024647620826184323, 9.270509076169233883855258397314, 9.348488303049488144379054139012
